Registering individual airframes doesn't make much sense - it's not unusual for a single quadcopter to go through many design iterations over the course of a few months.

It might start out as a 10" wide quadcopter, until the first major crash. Instead of ordering a new arm, it can be reconfigured as a tricopter with the addition of a servo. After a while, the tricopter might become a V-Tail if the user likes the flight characteristics.

Then FPV kit might get added, or a GPS reciever. Now it's flying semi-autonomously and the user wants better quality video, so it's transplanted onto a 18" frame, fitted with a gimbal and nicer camera.

At what point does the multicopter need to change registration numbers?

A simple "You must have your name and contact info on it" rule would be more than sufficient - and in fact is already the case for 90% of the aircraft out there. I even have my contact info taped to the body of my little 4" Hubsan quadcopter; I'll probably never need it, but why throw away expensive hardware if you don't have to?

It's not clear how one can get the serial number from the drone. ADS-B might be an option not only for tracking violators but also to solve "see-and-avoid" problem too.