By creating a system whereby everybody can be assumed to have a certain unique identifier, they put people in a situation where they can be made to divulge that number of various private purposes.

For instance, if a drivers' license didn't have a unique number, then stores couldn't be tempted to demand said number when processing a return.

It's akin to the difference between naive full disk encryption, and a true steganographic filesystem.

There's zero reason that SSN per entity shouldn't be unique, official use only, expire, etc.

Currently, SSN reveal detailed personal history when used by someone who knows how to read the encoding, which is public information.

Beyond that, use of SSN for non-official use is illegal, but the government doesn't do enforcement.