Ridiculous how easy it is...the feds could crack down today if they wanted to. With databases like Experian or Equifax, you can see how many multiple socials there are & who they stole them from.
Hell, if they spent 1/10th the effort they spend on watching US, this place would be tighter than a drum.
Easier than you said! We discussed this on FR many years ago. There is a SS Master File that lists all SS accounts; it is bounced against the IRS filings list. Mismatches are notified to correct their information. By law, neither the IRS nor the SS Administration are allowed to share that data with law enforcement (ICE or C&BP).
It wouldn't take much work to craft a bill to remedy that situation; other than the will to do the right thing.