It was pretty cool as a kid because you could use the Xbox 360's Kinect peripheral to pet them and teach them tricks, even if it was kinda just that iirc!
There was a version of nintendogs that was cats and catz! game i believe that was similar in concept! I'm pretty sure the Switch version of nintendogs has cats too.
Ye! I have the Petz Dogz game and loved it a lot! It was pretty similar to nintendogs but the dogs wouldn't run away and you didn't have to play every 8 hrs to keep them happy. The Horsez one was great too but my DS screen was a bit busted so I never got past a certain point.
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Catlateral Damage, Cattails (Wildwood story), Calico, Snacko, perhaps even Shelter 2 come to mind too.
If you want to reaaally stretch it, A Hat in Time has a mafia cat dlc..
Not played any of the others mentioned ^^
If you have a PS4, you can get a REALLY big kitty too in form of Trico (The Last Guardian)
Or uh.. at least, cat-like. Still can recommend!