Why not just allow you to take a picture with the front facing camera? Simpler, will work with a modern iPhone, photographic evidence of the person who has your phone.
The only downside I could see to this would be people freaking out because it would seem like Facebook is "tracking" them on their phones all the time. But I think it would be super useful if you did ever have your phone stolen.
This would effectively give you the full identity of whoever stole your phone
Not necessarily. Many people have fake names, etc on Facebook.