That would be very illegal. It looks harmless on the surface, but it's actually a big deal.
When people play the address game, they're thinking it's an easy way to save a few bucks on taxes and / or insurance premiums. But they're not thinking about who else is known for playing the address game, or how serious it is.
How serious is it? Your registration and insurance documents are a form of identification, and must be shown to law enforcement upon request. If you show a cop your registration card with a bogus address, you just produced false evidence to a government official. That's a criminal offence in most jurisdictions.
And who else is famous for using false / multiple addresses, you ask? International terrorists, organized crime bosses, cartel smugglers, fugitives from the law, scam artists and human traffickers come to mind. And believe me, you don't even want to be slightly suspected of being any of the above.