Why is dog licensing that crazy? You are required to license and register a car that's used in public. Your dog may always be on your property but for almost every other dog that is not that case.
A dog can be a public health threat, in both terms of disease and attacks on people or other dogs. Many states require rabies vaccinations for example .The owners are ultimately liable for that and there should be a way to track that.
Plus the fees frequently fund animal control and shelters in many areas.