Considering that cats are the original parkour artists, it has always seemed to me that making a game where you play as one should be a no-brainer, finding unlikely places on which to balance and using that vertical leap to great advantage.
I neglected a more primary characteristic, however - wantonly knocking shit over, which is the goal of Catlateral Damage, where you score points by pushing as many things off the shelf as possible. And because you're a cat, you don't just totally get away with it, but you also get showered in affection by way of points. The game is purrfectly free to play in your browser - check out Kotaku's demo video below first if you'd like.
For added realism, it should let you chase flies while getting bonus points for falling objects - sort of like catching the golden snitch in Quidditch. Or, hell, make it an invisible fly - if my cat is any indication, the bug can be completely imaginary, and it is still subject to chase.
source: Kotaku via SlyDante7777