If only there were some sort of moniker you went by other than a SSN. That coupled with some sort of location data could narrow you down to, ooh, one person I suppose.
In the UK we have NI number, which is used by my employer, pension provider, the tax services, the benefits office, and that's pretty much it. It's not used as a general source of ID by corporations.
From an outside point of view, the US system looks crazy. And that's coming from somebody who's country has a TV license.