But they still have id cards and registering with police when you move :-(

With a county office, not the police.

It doesn't matter too much since most people are somewhere "in the system" anyway: pretty much everyone is registered with various social security systems, employed people are known through income tax handling, etc.

(We also have rather strict rules on name changes and other means that could allow someone to make themselves disappear)

there are similar mechanisms in the US . in many cities, a landlord is required to register a tenant. similarly, purchase of real property is public record...

Tenant could be an LLC registered in a jurisdiction that doesn't require disclosure of ownership or directors.