Public universities are state agencies. What they require to issue ID and if anyone accepts it are different questions. But state issued photo ID isn't an unusual standard.
Form I-9 says school ID card with a photograph is a document that establishes identity.[1] It doesn't say it has to be state issued.
Form W-2 is a wage and tax statement from your employer. What identification is involved?
[1] https://www.uscis.gov/i-9-central/form-i-9-acceptable-docume...