I suppose it is somewhat common to use the name itself as part of the description of a character's overall design. I just used the name "Mobius the Raptor" for the nearly the whole first year I was around before updating to "Mobius Whitetalon" which still describes the talons and claws being white in color.

My only hypothesis is that in a way a name is used as a capsule description of a character until one can visually or more extensively describe the character. I don't know if anyone who does that intentionally does so, though. You can say, "My name is WoLf2Tails" and one would imagine a wolf with two tails as opposed to a wolf with one tail or a wolf with 9 tails. Or I could say "My name is Mobius Whitetalon" and someone would imagine someone with talons that are white, rather than black or green or some other color. The name helps identify WHAT is unique to us as opposed to a character designed by someone else. True, not everyone does that, but it doesn't seem uncommon either.

My primary has my animal in it

but Adara my brown hyena does not
Azora my wolf does not either

Well, if you count the species name "Chakat" in my name, then it would be a yes.

Well Romani does not.

Long Xian is not fur sure. Long Xian is Pinyin (Mandarin one language of Chinese) spelled with the Latin Alphabet for "Dragon Mountain." Long Xian is a Giant Panda.
Skylar Jaeger doesn't. He is a Husky
Petrus, My Sugar Glider doesn't
Bruno, My beaver costume, doesn't.

It was just a thing I threw together for Sylverwolfe. My father asked me what I wanted to be my email address when he set the family's emails up. I quickly thought of two things I really like. Silver coins, Notably US Silver Dollars, and Wolves, hence the name, The fact it described the species of my character was not something I consciously thought out when I came up with the name.

Nope. Not a wolf... Well, sorta. xD