The SSN according to law is supposed to be only used for SS purposes.
Been in the military? Been to college??

It used to be very important to keep it private. Now if someone wants it they can get it.

As far as this person requiring these kids to have it, it's just another underpaid retail clerk who doesn't know their job real well. Not surprising.
I had a friend one time that couldn't buy .22 bullets at Fred Meyer because he was 20 and you have to be 21 to buy pistol ammo. The dumb woman behind the counter insisted that they were pistol ammo. Go figure that one out. I guess they could be, but 90% od .22 bullets go into rifles, actually more like 99% probly.
He argued and argued and finally gave up. Drove down the street and bought them.