This app was mentioned in 2 comments, with an average of 1.00 upvote
It would help to state your OS, so we can filter out titles that are exclusive to the OS, but most of them should be for iOS/Android these days. Here are a few that I know of, but not exclusive to dogs or cats:
Clusterduck
Hamster Life || ANDROID || App Store
My Singing Monsters
Flutter Butterly Sanctuary
Dragonvale
Pocket Frogs
Tiny Farm || ANDROID || App Store