Yes, some places have "dog license". They are for different purposes than we are discussing in this thread. They either try to limit the number of dogs per family, or trying to keep eyes on whether or not your dogs have had vaccine. None of its purposes has anything to do with the requirement of be able to take care of the pets.
Pets are dying and abandoning in millions every year because of irresponsible pet owners. Have a license agreement to make sure everyone not only has the knowledge to take care of the particular pet species but also will never ditch the pet is the best way to solve the root problem.