Lets see... one of the characters in Spirited Away was actually a dragon, and there were lots of people that got turned in to pigs. I'm not sure if demons count as animals (well, I just counted a dragon too, but that's something people once thought was an animal.), however there were also quite a few demons in it.
Yu Yu Hackashu (sp IS wrong) comes to mind as well, most of the demons in it would could be considered animals or at least different.
Besides those you have pets and things from various anime... but I don't think that's what you were looking for.
A counterpoint, thinking of demons, might be Digimon. I'm sure all those monster/animal things were based on some portion of Japanese culture, some (many?) based on their mythology, but others based on the more modern perception. It may also be helpful to see how various animals/mythologies are represented in various genera of Japanese animation/manga. Everything from little boys/girls up through older dudes and ladies.