I'm pretty sure when I lived in the US for a couple of months I had to register with the municipality, just like here, so surely they should know.
Register with the municipality? Not sure what you're smoking. This isn't, e.g., Germany, where I'm told you have to do this within days of arrival.
In my whole life I've never had to proactively register with the town or city that I moved to.
The state I live in now does require that the towns take an annual census, so eventually the town finds out about you. Also if I want to vote – and I do want to vote so... – I have to go to town hall (or someplace similar) to register to vote. So that's another way the municipality knows about you.
If one wants their children to attend the public schools, they have to provide proof that theyI live there. Often that's done with a utility bill, e.g. electric, water, or gas bill. Or some other proof of residency.