Behind the scenes
The cat held by Marlon Brando in the opening scene was a stray Francis Ford Coppola found while on the lot at Paramount. The cat was not originally called for in the script, and the cat with its purring muffled some of Brando's dialogue, and, as a result, most of his lines had to be looped.
The use of the cat in the scene has later been interpreted as representing the hidden claws beneath the Don's warm facade.
In the video game
Vito's cat wasn't exactly seen in the Godfather game, but he can be heard at Corleone mall meowing, or the player can also hear sometimes a the mobsters saying "Sounds like the cats are fighting" which is probably referring to Vito's cat.