SSN's in the US are not unique, though they are only used by one person at a time.
I know someone with misspellings of their name on one bureau's credit report and its never caused a problem (he's too lazy to ask them to fix it). The misspelling is listed as his name and the right spelling is listed as "other name" (or something like that).
SSN and ITIN: nnn-nn-nnnn EIN: nn-nnnnnnn
So it covers not only all the individual taxpayers but also all the employers and businesses that are taxable separately from individual income tax payers.
However, there are a few reasons someone will get a new SSN number. Witness protection for example.