Pet owners should be required to obtain licenses for their cats, dogs, birds, and all kinds of exotic animals and should be required to renew them every year or two. Common animals like cats and dogs should be required to get sterilized (spayed/neutered) or pay an additional annual fee for each year in which it continues not being neutered. Exotic, unusually large, or notably dangerous pets should require a special license and payment of an additional fee. Common fish, unless they are unusually large or pose a significant threat to any nearby ecosystem if released, should not require licensing.
As a condition of licensing any pet, pictures and DNA samples of these animals should be taken and kept on file with the animal’s registration/licensing papers so that a connection between an owner and their pet could be made in the event the pet ever became lost or for whatever other reason a connection needs to be made. This DNA record would also help the government identify and prosecute pet owners who may have illegally released their pets (such as snakes, alligators, etc.) into the wild or local ecosystem.