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.
There is a commercial center here that sells exotic animals like snakes, rare birds and even insects like scorpions and spiders. There is a law against selling those animals but obviously the commercial center is protected by the authorities. Aside from live animals, there is a store that sells stuffed animals and dried mostly used for medicinal purposes like dried sea turtle and pangolin, the endangered specie of ant eater.
On a brighter note, one trafficker of banned animals was caught red-handed when his house in Cavite - just 20 kilometers from Manila - was raided by the cops. It yielded countless of exotic birds and albino snakes of all sizes. And although the guy denied the allegation, the raid was documented by media so there's no loophole for the trafficker to escape the case. The Department of Environment and Natural Resources official said that those animals are banned because they are endangered.