Well even if they have various names they should not have various SS numbers.
I am not naïve enough to believe it doesn’t happen but even so it is no reason not to pursue a relatively simple procedure to reduce the fraud in the system.
Eventually the criminals will work around this but you have to do what you can do.
My wife used to work the receiving desk at the ER. Patients or relatives would pull out identity after identity until one of them worked getting them into the system.