If I had to guess, the registry is such that if an elf is 'stolen' from another owner, then it would call into question the ownership of any other elf present in a location. Meaning that even if the other elves were actually legally owned, then any owner who has a 'stolen' elf in their possession... would in turn lose ownership of any legally owned elves as well.
So, by using stolen identities, it's more a cover for the owners than the elves; as I doubt the owner would be held criminally or civilly responsible if they weren't aware that the elf they had just 'purchased' wasn't who they claim to be. A kind of plausible deniability that doesn't draw attention to the others present, and also doesn't stop the owner from trying to help other elves.