This explains more. This short, simple and easy to understand page describes what makes a person illegal.
"SSA will not assign an SSN to an alien currently in lawful immigration status in the U.S. who does not have DHS work authorization unless the alien has a valid nonwork reason for an SSN."
https://s044a90.ssa.gov/apps10/poms.nsf/lnx/0100203510?opendocument
Yes, plain as mud! It goes on to say that none of the "bennies" described therein are good enough reasons for them to bother with the paperwork to give somebody a SSN, so "just do without" seems to be the order of the day. They're getting them somehow.
I never knew some of those existed - like temporary assistance for needy families. I assume single people are not worth helping. But I really didn't know the gov't helped anybody with a program like that.