At least in the US, all states have some or the other ordinances and regulations around pets. Not only are you required to register your pets, but there’s limits to what you can keep as a pet and how many. Otherwise it’s not hard to see at least one person in the neighborhood hoarding 20 dogs.

Every city in Ohio for one.

https://codes.ohio.gov/ohio-revised-code/section-955.01

Out of interest, why does it sound kind of crazy?

Here, pets have always had to be identifiable: historically with a collar, but microchips have been required for some years now as a more effective method.

(That applies nationally, not some city thing)

All 50 states have dog licensing in some form, but it seems that it's more a case of the state allowing municipalities to do so or not. Some states also require veterinarians to report unlicensed dogs. Larger cities almost certainly require it (NYC and Chicago definitely do).

https://www.rover.com/blog/dog-license/

Usually it's an animal welfare and control thing: you pay a lot more if you don't spay/neuter, rabies vaccination is required, etc.. Otherwise it's too easy for a city to be overrun by strays.

Also, if your pet gets lost and ends up being picked up by animal control, it's considered a stray if it's not registered.

If I remember correctly, EU will make chip tracking for pets mandatory by 2030 to unify laws that are currently made by individual states. France had this mandatory for over a decade.

If you want to travel within EU with your pet, you'll need a certificate for that as well.

No idea about how it is is the US but that doesn't sound crazy to register pets as they are at risk of being lost, abandonned, lacking vaccination or vets visits.