I suggested many moons ago that ship registration should be broken up by major power. Not for allegiances or anything but simply as a way to organize ship registrations and allow for some duplication of ship's names. A CMDR would choose between registering his or her ship in the Federation, the Empire or the Alliance. Within the major powers a ship's name could be used exactly once. If that name is not available in (for instance) the Federation the CMDR might try and register the same ship's name in the Empire or in the Alliance.
This way, each name can be registered in duplicate exactly three (3) times, once in the Empire, once in the Federation and once in Alliance registrations. Name-plates would display a small (major-power) sigil next to the ship's name so CMDRs would know which of the 3 possible ships of that name they are looking at. Of course, the first CMDR to register a unique name would have their choice of which power to register their ship's name.
I think you're confusing names with IDs. The ID is how the government keeps track of your ship, and needs to be unique. Usually they could care less about the name, and there's no good reason to restrict duplicates. I'm pretty sure there is already a unique value for each ship in the FD database, but it's probably just some ugly number.
If one wanted to get all complicated 'n stuff, you could have a VIN that is provided by the manufacturer and is supposed to be unique and unalterable. Nobody would be able to see it from outside the ship. ID tags (license plates) would register your VIN with some authority and provide a display to the outside world "Be on the lookout, Black & White Faulcon Cobra Mk III, ID Tag BDR529." Then a name you paint on the side that can be whatever you want.
The VIN would be unalterable by legal means, the ID can be changed by filling out the correct form for vanity plates, or registering in a different authority region and you can just repaint the name whenever you want.
Ooh, now I want to get COX049 vanity plates for my sidewinder.