The first time I saw how my understanding of identity is different was when I first talked with someone who was a "therian" as well (but didn't use the label for themself). At that time, I was still feeling closer to foxes and mentioned how being one was part of my identity. They responded that it was their identity, not just a part of it. That seems to be true for a lot of people, as far as I can tell.
I could never understand it because there's so much more that makes me "me", independent from the desire to be an animal. Like I said above, I love drawing and programming, I listen to certain types of music and watch anime, I'm quiet and introspective, I'm asexual and don't care about gender, I hate standing out, and so on.
There's just one identity, and that's "me" — the sum of all these things.
Volpeon (Bonus) 
@volpi@icy.wyvern.rip