So only public universities count, but not private ones? School ID's are not considered valid identification for any other government form, (I-9, W2, etc), so I don't think it makes sense to say they count here as well.
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...