Next Celsus assumes that the Jew (or Celsus) is making use of Matthew but this isn't necessarily the case. He goes to admit that he only assumed that Jesus lived in Egypt as an adult:
But, moreover, taking the history, contained in the Gospel according to Matthew, of our Lord's descent into Egypt, he refuses to believe the miraculous circumstances attending it, viz., either that the angel gave the divine intimation, or that our Lord's quitting Judea and residing in Egypt was an event of any significance; but he invents something altogether different, admitting somehow the miraculous works done by Jesus, by means of which He induced the multitude to follow Him as the Christ. And yet he desires to throw discredit on them, as being done by help of magic and not by divine power; for he asserts that he (Jesus), having been brought up as an illegitimate child, and having served for hire in Egypt, and then coming to the knowledge of certain miraculous powers, returned from thence to his own country, and by means of those powers proclaimed himself a god. Now I do not understand how a magician should exert himself to teach a doctrine which persuades us always to act as if God were to judge every man for his deeds.
The Jew clearly has a sophisticated literary ability but that doesn't prove he is one and the same with Celsus:
I do not think it necessary to grapple with an argument advanced not in a serious but in a scoffing spirit, such as the following: If the mother of Jesus was beautiful, then the god whose nature is not to love a corruptible body, had intercourse with her because she was beautiful; or, It was improbable that the god would entertain a passion for her, because she was neither rich nor of royal rank, seeing no one, even of her neighbours, knew her. And it is in the same scoffing spirit that he adds: When hated by her husband, and turned out of doors, she was not saved by divine power, nor was her story believed. Such things, he says, have no connection with the kingdom of heaven. In what respect does such language differ from that of those who pour abuse on others on the public streets, and whose words are unworthy of any serious attention?
Interestingly again Origen goes on to admit that the Jew seems to have had an unusual gospel - maybe not canonical Matthew:
After these assertions, he takes from the Gospel of Matthew, and perhaps also from the other Gospels, the account of the dove alighting upon our Saviour at His baptism by John, and desires to throw discredit upon the statement, alleging that the narrative is a fiction. Having completely disposed, as he imagined, of the story of our Lord's birth from a virgin, he does not proceed to deal in an orderly manner with the accounts that follow it; since passion and hatred observe no order, but angry and vindictive men slander those whom they hate, as the feeling comes upon them, being prevented by their passion from arranging their accusations on a careful and orderly plan. For if he had observed a proper arrangement, he would have taken up the Gospel, and, with the view of assailing it, would. have objected to the first narrative, then passed on to the second, and so on to the others. But now, after the birth from a virgin, this Celsus, who professes to be acquainted with all our history, attacks the account of the appearance of the Holy Spirit in the form of a dove at the baptism. He then, after that, tries to throw discredit upon the prediction that our Lord was to come into the world. In the next place, he runs away to what immediately follows the narrative of the birth of Jesus— the account of the star, and of the wise men who came from the east to worship the child. And you yourself may find, if you take the trouble, many confused statements made by Celsus throughout his whole book; so that even in this account he may, by those who know how to observe and require an orderly method of arrangement, be convicted of great rashness and boasting, in having inscribed upon his work the title of A True Discourse,— a thing which is never done by a learned philosopher.
The idea that the passages weren't in the same order as Matthew or any other canonical gospel is not an argument against the Jew or Celsus!
Yet most interesting of all, the Jew knows about the dove appearing as John the Baptist was baptizing him but assumes that John suffered a similar death to Jesus - all of which Origen finds odd for a Jew to admit. First:
But, that we may not have the appearance of intentionally passing by his charges through inability to refute them, we have resolved to answer each one of them separately according to our ability, attending not to the connection and sequence of the nature of the things themselves, but to the arrangement of the subjects as they occur in this book. Let us therefore notice what he has to say by way of impugning the bodily appearance of the Holy Spirit to our Saviour in the form of a dove. And it is a Jew who addresses the following language to Him whom we acknowledge to be our Lord Jesus: When you were bathing, says the Jew, beside John, you say that what had the appearance of a bird from the air alighted upon you. And then this same Jew of his, continuing his interrogations, asks, What credible witness beheld this appearance? Or who heard a voice from heaven declaring you to be the Son of God? What proof is there of it, save your own assertion, and the statement of another of those individuals who have been punished along with you?
Then Origen attempts to raise questions about a Jew raising the objections - because they should acknowledge the Holy Spirit - an extremely weak objection:
We shall therefore say, in the first place, that if he who disbelieves the appearance of the Holy Spirit in the form of a dove had been described as an Epicurean, or a follower of Democritus, or a Peripatetic, the statement would have been in keeping with the character of such an objector. But now even this Celsus, wisest of all men, did not perceive that it is to a Jew, who believes more incredible things contained in the writings of the prophets than the narrative of the appearance of the dove, that he attributes such an objection! For one might say to the Jew, when expressing his disbelief of the appearance, and thinking to assail it as a fiction, How are you able to prove, sir, that the Lord spoke to Adam, or to Eve, or to Cain, or to Noah, or to Abraham, or to Isaac, or to Jacob, those words which He is recorded to have spoken to these men? And, to compare history with history, I would say to the Jew, Even your own Ezekiel writes, saying, 'The heavens were opened, and I saw a vision of God.' After relating which, he adds, 'This was the appearance of the likeness of the glory of the Lord; and He said to me,' etc. Now, if what is related of Jesus be false, since we cannot, as you suppose, clearly prove it to be true, it being seen or heard by Himself alone, and, as you appear to have observed, also by one of those who were punished, why should we not rather say that Ezekiel also was dealing in the marvellous when he said, The heavens were opened, etc.? Nay, even Isaiah asserts, I saw the Lord of hosts sitting on a throne, high and lifted up; and the seraphim stood round about it: the one had six wings, and the other had six wings. How can we tell whether he really saw them or not? Now, O Jew, you have believed these visions to be true, and to have been not only shown to the prophet by a diviner Spirit, but also to have been both spoken and recorded by the same.
Origen is clearly here stretching the boundaries of credibility. A Jew might well object to a dove appearing as the Holy Spirit and the heavens opening up etc. And again:
And he who is adorned with the spiritual gift, called the word of wisdom, will explain also the reason of the heavens opening, and the dove appearing, and why the Holy Spirit appeared to Jesus in the form of no other living thing than that of a dove. But our present subject does not require us to explain this, our purpose being to show that Celsus displayed no sound judgment in representing a Jew as disbelieving, on such grounds, a fact which has greater probability in its favour than many events in which he firmly reposes confidence.
But then Origen immediately goes on to say that he debated with Jews in the exact or similar to the manner described. Desperate arguments. And again:
And although Celsus, or the Jew whom he has introduced, may treat with mockery what I am going to say, I shall say it nevertheless—that many have been converted to Christianity as if against their will, some sort of spirit having suddenly transformed their minds from a hatred of the doctrine to a readiness to die in its defence, and having appeared to them either in a waking vision or a dream of the night. Many such instances have we known, which, if we were to commit to writing, although they were seen and witnessed by ourselves, we should afford great occasion for ridicule to unbelievers, who would imagine that we, like those whom they suppose to have invented such things, had ourselves also done the same. But God is witness of our conscientious desire, not by false statements, but by testimonies of different kinds, to establish the divinity of the doctrine of Jesus. And as it is a Jew who is perplexed about the account of the Holy Spirit having descended upon Jesus in the form of a dove, we would say to him, Sir, who is it that says in Isaiah, 'And now the Lord has sent me and His Spirit.' In which sentence, as the meaning is doubtful— viz., whether the Father and the Holy Spirit sent Jesus, or the Father sent both Christ and the Holy Spirit— the latter is correct. For, because the Saviour was sent, afterwards the Holy Spirit was sent also, that the prediction of the prophet might be fulfilled; and as it was necessary that the fulfilment of the prophecy should be known to posterity, the disciples of Jesus for that reason committed the result to writing.
I see no reason to doubt the Jew as being a separate person from Celsus based on any of this.
But then Origen does bring up something worth considering - something which immediately follows his citation of 'debates' he's had with Jews. Jews know nothing about the John the Baptist:
I would like to say to Celsus, who represents the Jew as accepting somehow John as a Baptist, who baptized Jesus, that the existence of John the Baptist, baptizing for the remission of sins, is related by one who lived no great length of time after John and Jesus. For in the 18th book of his Antiquities of the Jews, Josephus bears witness to John as having been a Baptist, and as promising purification to those who underwent the rite. Now this writer, although not believing in Jesus as the Christ, in seeking after the cause of the fall of Jerusalem and the destruction of the temple, whereas he ought to have said that the conspiracy against Jesus was the cause of these calamities befalling the people, since they put to death Christ, who was a prophet, says nevertheless— being, although against his will, not far from the truth— that these disasters happened to the Jews as a punishment for the death of James the Just, who was a brother of Jesus (called Christ),— the Jews having put him to death, although he was a man most distinguished for his justice. Paul, a genuine disciple of Jesus, says that he regarded this James as a brother of the Lord, not so much on account of their relationship by blood, or of their being brought up together, as because of his virtue and doctrine. If, then, he says that it was on account of James that the desolation of Jerusalem was made to overtake the Jews, how should it not be more in accordance with reason to say that it happened on account (of the death) of Jesus Christ, of whose divinity so many Churches are witnesses, composed of those who have been convened from a flood of sins, and who have joined themselves to the Creator, and who refer all their actions to His good pleasure.
And then again:
And as it is a Jew who, in the work of Celsus, uses the language to Jesus regarding the appearance of the Holy Spirit in the form of a dove, This is your own testimony, unsupported save by one of those who were sharers of your punishment, whom you adduce, it is necessary for us to show him that such a statement is not appropriately placed in the mouth of a Jew. For the Jews do not connect John with Jesus, nor the punishment of John with that of Christ. And by this instance, this man who boasts of universal knowledge is convicted of not knowing what words he ought to ascribe to a Jew engaged in a disputation with Jesus.
Origen attempts to turn this around into an argument that the Jew identified 'John the Baptist' as the one meant to have 'shared the sufferings of Jesus' but I don't think is the case. If we go back to the original statement it is clear that there are two different ideas being represented here:
Let us therefore notice what he has to say by way of impugning the bodily appearance of the Holy Spirit to our Saviour in the form of a dove. And it is a Jew who addresses the following language to Him whom we acknowledge to be our Lord Jesus: When you were bathing, says the Jew, beside John, you say that what had the appearance of a bird from the air alighted upon you. And then this same Jew of his, continuing his interrogations, asks, What credible witness beheld this appearance? Or who heard a voice from heaven declaring you to be the Son of God? What proof is there of it, save your own assertion, and the statement of another of those individuals who have been punished along with you?
In other words, the Jew says to Jesus - while you were bathing beside John you say (i.e. Jesus) something which had the appearance of a bird landed on you - a second person acknowledges it who belongs to 'those individuals who have been punished along with you' i.e. a gospel writer not necessarily (and unlikely) John the baptist. Indeed the individual is never explicitly identified as 'John the Baptist' only John FWIW.