In Sweden your social security number is public information. What's the reason for it being private in the US? Sounds like a horrible thing to rely on for security anyway.
In the US, a combination of personal information is taken by most as proof that you are who you claim to be. If you know enough of someone else's information, you can steal money rather easily. We need a better identity solution, but it's what we have right now.
For what it's worth, here in Europe you could pretend to be someone else as well, if you have enough information, but what's the point when you can't touch their money?