Cats yes, you can get the “farmer” cats that are happy little accidents of no particular breed. They are not in the stores though. Neither are the purebred cats, which are similar to dog breeders.

Dogs are bought directly from breeders (who almost universally has it a a side hobby) and while not illegal to breed on known hereditary diseases any responsible breeder have various veterinarian’s certificates for (lack of) hereditary issues for the breed and puppies are checked by a vet before delivery.