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.
But if you change your address using a US postal service card it says that they are required to notify the IRS-of your address change.
So Postal service shares information with the IRS about your address but there can be no sharing of info with Police, ICE, C & BP)....