Yes, the Privacy Act is just one more problem associated w/ SSNs, but there are plenty of situations where people want them in the table anyway.
Theroetically, there is only one per person, but with 9 digits, 3 of which have geographic encoding, you can see where the government will run out of unique numbers pretty quickly, especially since some numbers (000-00-0000, 123-45-6789, etc) will not be used.
This is why you hear news stories about people who want to check on their Social Security benefit and find out that they've been dead for several years!
The situation is further complicated when you have the possibility of foreign nationals who have no SSN; now you have to dummy up a unique identifier for them!