They're not reliably unique: a nontrivial percentage of the population has multiple numbers associated with them, and a nontrivial percentage of the numbers are associated with multiple people.

If you have two systems and you want to correlate data between them, you can use something like the SSN to assist in that process, but usually it will need additional information.

Generally you can't use something someone supplies as a shared identifier. You can't start the process of verifying it, after all, without an identifier. You might think of all sorts of ways around this, but they all boil down to either tolerating errors or making the SSN a secondary attribute rather than a bona fide identifier.