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.
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.
Is it really the government doing this? It seems to me that it's businesses, not the government, that insist on using SSNs as consumer identifiers. While this drives me crazy, I'm not sure that it's reasonable to pin it on the government (which, essentially by definition, is the party that is intended to use them as identifiers).
EDIT: What delinka (https://news.ycombinator.com/item?id=12425976) said.
And the private businesses security help, such as Google's, only talks about government and not business attackers.