The real reason is farming and phytosanitary needs. The animals used need to be cheap enough to make sense to feed a pet, and farmed in a way where you can be sure your product won't give cats worms or other parasites. Cats also don't really eat the hair and hide, or feathers, of animals, so you ultimately have mammal, birds, and fish as categories. We already farm mammals, birds, and fish for humans, so the systems are already there.
That being said, I used to feed my little house panther raw food, which was a little frozen gore nugget of rabbit or chicken or duck. She loved them. Loved, loved, loved them. She's caught pigeons and rats before, so it's really not that far off. It's not like either would present vastly different nutrition as compared to a mouse or a rat or song bird anyway.

