Back when I started college, my SSN was my student ID number. It felt weird, of course. I think there was a change in the law soon after I started college, because it did soon get changed into a different number of the same length.
Later on, I did a brief stint working for the federal gov't. In that setting, they used the SSN as our employee IDs. It was on all the personnel forms, and often seen on "list of people in the department" spreadsheets. Of course in order to comply with some law, these forms would also have a footnote explaining why they needed the SSN.
From these experiences, I have a very hard time actually thinking of the SSN as the sort of "secure password" everyone else wants to insist that it is. Unfortunately, I'm not aware of an alternative.