Non-indigenous is to vague a definition for exotic animals. If we were to get rid of all non-indigenous animals from say America, we would have to get rid of all horses, most cows species, some pig species, most dogs, most cats, and most pet birds, and that's not including the arthropods. Heck most of our pets were domesticated in either Europe or Asia.
The law should be more specific about the types of animals you can't have as a pet. I understand people not wanting their neighbors to own lions, bears, tigers, wolves, and other huge predators that might get out when their owners can't afford to feed them. I also understand some reptiles being forbidden because of the problems that places like Florida are having with crocodiles being released close to residential areas and are unafraid of people or the pythons that owners are carelessly releasing into the wild. But some pets, like hedgehogs and chinchillas aren't going to grow up into monsters and eat you and your children. The law should simply require that you neuter them so that if they get out in the wild, they can't reproduce. Then have you get a special license to have them that can only be renewed with proof of shots, health, and neutering.
