I strongly feel that a license should be mandatory. A lot of people get pets because they saw someone else do it, and don't know how to properly take care of it. There are also differences between similar kind of animals that pet owners should know, but often don't. Owning a dog doesn't make you prepared to own a cat, and there are differences even between different breeds of the same race. So, there should be law-enforced classes and licenses for every pet owner. Along with the license, stronger pet protection laws should be enforced too, otherwise there's no point. Of course, they will always be people who mistreat their pets and breed them illegally, but with proper punishment a lot of people will be discouraged. The license would also serve to weed out people who wouldn't even pay the license fee to own a pet. Honestly, I see nothing bad coming out of it, either it will have a positive impact or none at all.
About the fishing and hunting licenses, it's a bit funny how you need a license to kill some animals, yet you can torture others to death without any charge at all.