We love our cats and dogs, and some people collect other species: birds, snakes, horses, pigs, lizards, insects.
It seems to me however that one function, the MAJOR function of a pet is in the name: we need to touch.
Cats love to be petted, dogs love to be petted and scratched. We love to touch them -- there's a primeval pleasure we mammals derive from physically touching. Elephants are big touchers, dolphins are big touchers, and of course all primates are big touchers.
Reptiles not so much.
Birds seem to preen themselves and occasionally other birds, so maybe they're touchers too.
The name "pet" seems hard to apply to any animal outside the mammal kingdom.