He never says he or his hands are like a lion (ari), since his lexicon is always to use the word aryeh for lion.
Real answer is that you are right that it is poetry, a chiasm, which is common in Hebrew poems.
Look at this part:
But You, O Lord, do not be far from Me;
O My Strength, hasten to help Me!
Deliver Me from the sword,
My precious life from the power of the dog.
Save Me from the lion’s mouth
And from the horns of the wild oxen!
It's the bottom half of a chiasm because the bold ideas are in reverse order from the top half in describing the enemies' ability to attack him. Each part of the chiasm lists a method that a different type of enemy can use to attack.
Here is the top half of the chiasm:
Many bulls have surrounded Me;
Strong bulls of Bashan have encircled Me.
They gape at Me with their mouths,
Like a raging and roaring lion.
I am poured out like water,
And all My bones are out of joint;
My heart is like wax;
It has melted within Me.
My strength is dried up like a potsherd,
And My tongue clings to My jaws;
You have brought Me to the dust of death.
For dogs have surrounded Me;
The congregation of the wicked has enclosed Me.
They pierced[c] My hands and My feet;
You can match them up in the same order:
1. bulls or oxen horns
2. Lion's mouth
3. dogs surrounding
or dogs' power
4. SWORD [BOTTOM HALF of CHIASM] ________
[Top half of Chiasm]
So fill in the blank, what goes in this top half of the chiasm that fills in the blank and matches with sword?Pierced.
(Although I would better translate the LXX / Minority Masoretic Reading as "gouged")
In fact, the piercing of the narrator is the climax of the chasm of the list of the enemies' powers.
At the climactic point it gives basic indications that the narrator is killed like being lain in the dust of death, pouring out (what happens when someone is killed by gouging), the bones going out of joint (doesn't that happen during crucifixion?), the strength drying like a broken pot, the enemies dividing the clothes (that happens after someone is killed by enemies, they accomplish their goal and they take the nice clothes that the victim doesn't need.)
So you are right poetry is a big part of this. When you figure out how the poetry works, it shows it's about the narrator getting killed and resurrected.