SSNs have been used as student IDs by particularly stupid educational institutions. The 'nice' thing about getting SSNs from students is the likelihood they'll live for a long time after the breach and thus be subject to identity theft for many years to come.
This was common years (decades, really) ago but I'd be surprised if any university today was still doing that. I guess there could still be some....
My university stopped putting SSNs on student IDs more than 25 years ago. I'd be surprised if there are many who still do that.
Though I wouldn't be surprised if some 40 year old university IT system requires its use as an identifier, regardless of whether or not it gets printed anywhere.