SSN's are not unique, and were never intended to be. They are assigned using an allocation scheme that ensures people of similar age will not have the same SSN, and that allows localities to independently allocate them at birth without creating collisions.
[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.