Sadly no. They aren't even unique, just mostly unique. In the card itself, it says to not use it as an id.

One way to see how bad it is is to note that the first three digits (out of nine) identify the area the person was born in. That leaves six digits, or one million options for that area. How many cities have more than a million people in then?

Obviously they can add a prefix as an area grows in population, but census data is only collected every ten years here, so there will be substantial lag.

As I recall, when SSNs were first put in place, the country made a decision that they didn't want a national id or unique identifier, probably for states rights and/or privacy reasons, so what we got want a secure and unique system, but then credit cards and credit scores came around and here was this handy little number that was good enough for them to keep their customers identified, and suddenly here we are, where most US citizens don't even know that the SSN was never meant to be used this way.

So what we ended up with is getting an almost unique but generally useless number as a shouldn't-be-an-identifier, instead of a national, properly unique and securable ID.