There's already a topic for Tokyo Jungle so forget that. I add an emphasis on "as" as apposed to "with" animals, like Nintendogs or Endless Ocean.
I know of a game called Dog's Life for the PS2. I brought my copy several years ago when it was new.. It's an action-adventure game to play. You play as an American Foxhound named Jake. His love interest, Daisy, gets taken by some dog-catchers. You need to visit three areas and rescue her. In the game you can "take over" over fifteen different breeds but you'll mainly play as Jake.
There's a similar game called The Dog Island but I don't own that title yet. I know Petz Dogz 2 ad Petz Cats 2 also have you as animals but Petz Horsez 2 has you as a human. Arctic Tale is based off a movie and apparently you can play as a polar bear cub, fox, or walrus.