This is a good summary. I had to go pull up wikipedia on it since I roughly knew that social security was a national insurance/pension kind of system but am actually hazy on details.

The major issue with it as id (aside from DBA's gripes about it) is that credit agencies and banks started to rely on it for credit scores and loans. You see, the US has a social scoring system (what we always accuse China of) but the only thing it tracks is how reliable you are about paying off debts. So with your home address, name, and SSN, basically anyone can take out loans or credit cards in your name. This will then damage your credit score, making it harder to get loans, buy a home, rent property, or even get a job.

That's why Americans are always concerned about having our identity stolen: because you don't need a lot of info to financially ruin someone's life.