> Can you do that within any one single country?
I don't think so, this would be surprising.
> Nothing of that is about your citizenship, so what does it have to do with another country?
I am sorry I don't understand this point.
> In Germany I have to register within 4 weeks of moving anywhere, it has to be a place that is registered as residential zone and you need an owner-ship or rental agreement certificate (even when living with your parents).
For the record, in Belgium, when moving to a different administrative territory (for lack of a better word, think cities, small territories) you have to register to your new town hall and a police officer will check you actually moved in.
But if you are homeless and still have a social net the social worker can register a public place as your contact address so you keep your rights and their benefits.