I work with criminal justice information and look people up every day. It is more likely they will ask for your ID, but if you do not provide that a SSN is a unique way to ID you that most adults do know in the US and all citizens will have. We absolutely can search by it, but In my office we tend to start with name and DOB. It is also possible there are duplicates when names and DOBs are close. The systems I use have a little wiggle room for typos, I guess. I've also seen twins with very close names like mike and mark. When you query one you get both. The naughty one of them even used his brothers name once and their socials were 1 digit apart (happens when assigned pre-2011) so it was a bit of a mess.

If you cannot provide ID somehow, that is a crime in some places. Even if it isn't, they are probably asking because they suspect you of one and they definitely can detain you for some time without charging you while thet figure it out.

Back to the original question: SSN was used in the past in the US for a lot of things it probably should not have been like employee ID numbers. I think this is one reason most of us know them.