"You become responsible, forever, for what you have tamed."
- The Little Prince
Exotic animals are hard to describe but easy to identify when you see one. Most vets only see standard house pets: cats, dogs and maybe ferrets so to them an exotic animal is anything beyond that. Which is something to consider when owning an exotic animal, make sure you have a vet that will take care of them when they become sick or injured.
Although the phrase exotic in reference to animals simply could mean any non-native animal or species. However, I would imagine that most people would assume that a skunk, porcupine, raccoon, etc would be an exotic animal and those are native across the US. So now we have some further refining to do to thoroughly answer what an exotic animal is and is not. If we state that exotic is in reference to the "familiarity" of owning said pet than we can get a better idea of what an exotic animal is regardless of where the animal is native to. So in that instance an exotic animal is any animal that is typically not considered a domestic pet.
Some animals just don't make good pets and should not be domesticated. If the animal is naturally nocturnal and you want them as a pet than you should allow them to still live as nocturnal and not try to change them. Cats are nocturnal and it took generations of domestication to change that for cats, some are still very much nocturnal.
Climate is another consideration when talking about what an exotic animal is. If the animal needs appropriate living conditions, either heated or cooled facility than it could be an exotic animal. Remember that the animal will be living year-round in captivity, not just part of the year so make sure that you are able to provide them adequate living conditions.