Thirty years ago, you only got an SS# when you started work, but currently all American citizens get SS#'s assigned at birth, and that's been the case for a while.
Back when I was an employer I had to fill out a state form every quarter for unemployment compensation taxes, that listed the SS# of every employee and how many weeks they worked
All it takes is a central web site where employers log the employee-provided SS# and name. If the number is invalid, it kicks out immediately. If the same number is provided for too many different employers, in locations too far apart to make it seem likely that the same person is really working there, the employer gets a follow-up call. If too many hits occur against an employer, he gets a surprise visit from INS
The problem is you can have a real number but it isn't for that person. That is what I am referring to. All of these things mentioned are useful and are being used. But as it currently stands it is very easy for someone to slip through the holes.