not sure either but as you experienced (like me) we had them as our college identification numbers back in the day and so far as i know colleges still use them internally (b/c you have to provide them if applying for student aide or student loans). 'social security' as defined by the feds encompasses a myriad of federal. state and local programs so they have the legal right to collect and use the numbers-public assistance, snap, food and nutrition services (free or reduced school lunch, meals on wheels and dozens of other programs fall under this umbrella-all who have that same right to an ssn), housing assistance, energy assistance, earned income tax credit...use of it for driver's licenses is permitted by federal law (though a handful of states do not require it),
the privacy laws are very complex and some entities still have a legal right to collect the numbers b/c of VERY minor affliations with larger entities. it might be that they receive a tiny amount of money from another entity that long ago was (and in the fine print-still is) designated as 'public assistance', well that pops them under the umbrella of being allowed to collect the number. i worked for social services which decades earlier in that region had separated so that it was separate from 'health services' BUT both did'nt entirely sever themselves from each other b/c both had certain abilitites and access to records/data due to their definition by the feds/state and did'nt want to lose those.