SSNs are not unique for a whole lot of reasons. https://www.nbcnews.com/technolog/odds-someone-else-has-your...
[edit] There is a related claim (not made above) that is correct, which is SSNs were never intended to be a secret.
In the past, it was actually illegal to demand someone's SSN for non-official use. The social security administration seems to encourage SSN harvesting now. I'm not sure what changed. Probably scope creep / financial incentives leading to increasing privacy violations.