So you empathise with your pets whom you have a kind-of relationship with. But you eat e.g. pigs because you don't.

Counterpoint: in the UK, rabbits are fairly common pets and uncommon-but-not-taboo food. Chickens are sometimes kept as pets too, though I'm not sure whether that tends to be mostly for the eggs. Horses are accepted as both pets and food, though rare in both cases. I'd be surprised if dog-eating cultures don't also have pet dogs. I believe there are other cultures that commonly keep cows, goats and pigs as pets, and I'd be surprised if those cultures don't eat those animals.

(This isn't the same as someone eating their own pet, but no-eating-your-own-pet wouldn't explain cultural taboos against any forms of meat.)