My guess is it's more about liability - if a dog is involved in an incident, you can prove who the owner is and potentially hold them liable (depending on the circumstances, of course).
I would if it made sense, but it doesn't, unfortunately
Dogs are more dangerous than cats for humans, that's enough to justify why to own a cat you don't need a license while to own a 70 pounds dog you do.
I live in a country, Italy, where you don't need a license to own dogs, but I can clearly see why they aren't needed for cats where dog licensing is mandatory.
