Did we used to have dog licences? I think thats a good idea actually.
So do I. They used to be (thinks) something like 7s6d - 37½p, and you had to get them from the post office. Then some time in the 70s, someone realised they were getting about £1,000,000 from them, but spending £10,000,000 administering the system
Rather than up the cost to (say) £5/year, it was simple to abolish it.
Now for myself, I think that owning a dog - like driving a car - is not a god-given right, but a privilege. A properly enforced licensing system might just reinforce that, and encourage responsible dog ownership. The money raised from those that do pay it could be used for dog welfare in general (or at the very least to clean up the dog shit left behind).
Those that don't pay would have the dogs confiscated.
If so many people are happy to have a human ID card system, then a canine one should be a shoe-in.