I used to work as a database administrator in a community college.
We required a copy of those documents to cut down on mistakes. People often write down the wrong number on the application, or a wrong number gets copied from one form to the other.
So, we needed a "primary" source of data, namely the photocopy of the actual SS card and actual drivers license, which could be referred to if there was a discrepancy between different records. That happened fairly often, unfortunately. As the DBA I, personally, had to query the database for inconsistant SSNs and resolve any issues. I did that routine about once a month, and almost always referred to the original application photocopies for resolution.
As for the file cabinets being accessible to students, please believe me that in our office we had strict rules about who could so much as touch a file cabinet, or who could so much as touch a file. It made things difficult, as often we would hire student workers to assist us. A student worker could not touch certain files, so we employees had to retrieve and re-file things ourselves, something that a lower-paid non-managerial clerical worker would do in any other office. We all had lockable desk drawers and lockable offices, and any sensitive files were locked away when away from our desks, even when we were away for a few minutes.