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There can be duplicate SSNs due to name changes of an individual, that's the easiest answer. In general, it's common to just add a new record in cases where a person's information changes so you can retain the old record(s) and thus have a history for a person (look up Slowly Changing Dimensions (SCD)). That's how the SSA is able to figure out if a person changed their gender, they just look up that information using the same SSN and see if the gender in the new application is different from the old data.
Another accusation Elon made was that payments are going to people missing SSNs. The best explanation I have for that is that various state departments have their own on-premise databases and their own structure and design that do not necessarily mirror the federal master database. There are likely some databases where the SSN field is setup to accept strings only, since in real life, your SSN on your card actually has dashes, those dashes make the number into a string. If the SSN is stored as a string in a state database, then when it's brought over to the federal database (assuming the federal db is using a number field instead of text), there can be some data loss, resulting in a NULL.
JFC: married individuals, or divorced and name change back, would be totally fucked. Just on the very surface is his fuckery.
Hypothetically you could have a separate "previous names" table where you keep the previous names and on the main table you only keep the current name. There are a lot of ways to design a db to not unnecessarily duplicate SSNs, but without knowing the implementation it's hard to say how wrong Musk is. But it's obvious he doesn't know what he's talking about because we know that due to human error SSN-s are not unique and you can't enforce uniqueness on SSN-s without completely fucking up the system. Complaining about it the way he did indicates that he doesn't really understand why things are the way they are.