When you get past the Harpy quest, you are asked to tame a wolf. This wolf has a unique skin (as far as I can tell, as I don't see it on petopia) I don't recall ever seeing a wolf with bands available on it in the game.

There's also a named Elite on the Isle that has a re-used diseased bear skin, but retains its name when tamed:

Also, I know it doesn't help without the names, but I went through every single beast on the island and was able to cast Tame Beast on them. The only one I couldn't was Bloodbeak, but it was confirmed tameable earlier.