I had a conversation the other day where SSN was considered viable to replace using a password. Yes, it simplifies single sign-in, but it has major consequences (and there are other ways to do single sign-in). Using SSN as a password is a bad idea, the kind of thing that blows up in your face with security and functionality problems.
You can't change your SSN (unlike your password)
SSN is too limited. It's merely a nine digit number, so it can't even pass strong password tests. Furthermore, the last 4 digits are often available (so down to 5 digits left), and there are rules that an SSN must adhere too (they aren't just any random 9-digit numbers). With less than 50,000 possibilities, it's fair game for a brute force attack.
SSN is publicly available in many cases. For example, HR and managers can often see your SSN (but not your password).
SSN has direct real life significance too, and is therefore immediately dangerous if lost. For example, if you lose a password, that's bad, but it's only dangerous if the hacker (A) knows the context - i.e. the system to apply it to, and (B) the password hasn't changed yet. But the context for an SSN is immediately known (like any financial or government institution), and SSNs can't change.