As a software engineer I can assure you that this is absolute bullshit

People (fraudulently and/or accidentally) use multiple social security numbers.

https://www.nbcnews.com/technolog/odds-someone-else-has-your...

As people have pointed out it depends on the details. Maybe SSN is not the primary key and it's something else like name + DOB. It still requires more details to prove that it's being used maliciously but there's no reason to think it's completely b.s.

The problem here is that Musk's statement is meaningless drivel. That suits him fine because his purpose is not to convey information but to create an impression. The impression he wants to create is that the system is fundamentally broken and that only he, the great and wonderful Elon, can fix it. It does not matter for his purposes whether the statement is true or not. In fact, it suits his purposes best if, as in this case, the statement is neither true nor false because it is essentially meaningless.