Not OP, but sounds like the places in Germany where you go to get or extend your passport, register after moving to a new place etc. Notoriously hard to get a timely appointment for things that should not require an in person interaction in the first place.

As konha said, it's the government offices, providing services for registration, driver's licence, taxes, etc. It works the same way in several EU countries.

Sometimes you can just show up (and wait in a queue) but most often you make an appointment beforehand. In many states/cities, the waiting times are weeks and months long (even in my home city everything's booked a month ahead and you can see the big queues if you go in person).

People can indeed just refresh the page themselves and hope a spot opens up, but that's pretty annoying.