The SSN is technically a NUID, not an UUID. So legal names are more important in international context (an Indian SSN doesn't mean anything in the US, for example).
But yeah, assigning GUIDs to everyone born in the world, like as "a93sz0sz" would allow uniqueness for thousands of years.
Actually legal names are a problem even within the US. I recently had problems with getting a California driver's license because my second middle name was abbreviated on my green card, but spelled out fully on my North Carolina driver's license, which I was transferring to California.
But yeah, assigning GUIDs to everyone born in the world, like as "a93sz0sz" would allow uniqueness for thousands of years.