XD I don't know if you would call it a poem, but count the number of syllables in each line: it goes from one syllable to eight. I've also included notes on why I think each particular animal suits each particular character, and no, the animal might not be what you think it is. Be prepared for a zoology lesson too XD.
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Hatake Kakashi – Peregrine Falcons
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Prince
Black veiled
Piercing eyes
Wielding sharp tools
Lightning striking down
Far-ranging wanderer
Fierce guardian of its nest
Falter before its cold fury
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Prince - Back in the Middle Ages, when Western nobility commonly practiced falconry, peregrines were associated with princes in formal rankings of the birds of prey used. Kakashi, as the son of a legend of the Hidden Leaf, an elite jōnin in his own right, and all but slated to be the Sixth Hokage unless someone else steps up, can certainly be considered a prince of Konoha, in spirit if not in name.
Black veiled - If you've ever seen a picture of a peregrine, you know it has rather distinctive markings. If you've ever seen Kakashi, you know he has a rather iconic something.
Piercing eyes - Falcon. Sharingan. Need I say more?
Wielding sharp tools - The bird has a sharp beak and some wicked curved talons. The man is a dab hand with kunai, and I hear he's recently been sighted swinging a really big sword (I'll avoid detail for those who don't keep up to date with the manga).
Lightning striking down - With the peregrine, I mean this metaphorically, in reference to their characteristic hunting stoop, where they dive down on their prey from above at great speed, delivering a blow powerful enough to stun or kill the hapless pigeon instantly. With Kakashi, you know I mean this literally, what with his Raikiri, also known as the Lightning Blade.
Far-ranging wanderer - The name of the falcon means "pilgrim", in reference to their migratory habits. Kakashi isn't as bad as, say, Jiraiya, about his travelling, but the guy goes places too. He always comes back too, much like how the peregrine always returns to the same nesting site if possible.
Fierce guardian of its nest - Peregrines ferociously defend their nests against all manner of large avians and mammals; there have been cases of peregrines killing golden eagles in the name of protecting their nest, despite said eagles being large and strong enough to prey on the smaller falcons. We know how Kakashi is about looking out for his team; he's practically suicidal about it.
Falter before its cold fury - Peregrine falcons are badass. Hatake Kakashi is badass.