As far as I know TES had all unique animals until Oblivion introduced some our-world ones. I just loved Morrowind's Nix-hounds, Cliff Racers(although they were a pain), Netches(totally awesome), Guars, Kaggouti's. That's what I really love about Morrowind, it's really original.
Sounds like you never played Daggerfall (or Arena? I didn't play Arena, either). From the fighty things, we can add bats and tigers, at the very least. And then there are the sprite-only animals that populate the towns: dogs, cats, pigs, cows. I'm pretty sure there are camels in the Hammerfell side of the Bay.
Additionally, any time a TES race has both a specific name, and a descriptive name derived from an animal, you can presume the animal exists for reference. For example, Tscaeci and Immortal Snake Vampires: we can presume snakes exist.
I'm sure there are dozens of references to mundane animals in various in-game books, as well.