Are flight schools required to verify citizenship of all students?
You are required to verify the identity of people undergoing initial Sport Pilot, Recreational Pilot, Private Pilot, Instrument, and Multi Engine training.  Any other kind of dual instruction does not require verification of US citizenship or TSA approval of resident aliens.  See 49 CFR 1552.3(h).

The CFI must inspect the original documents and verify they are authentic.  The CFI must endorse the CFI’s own logbook with an endorsement per AC 61-65 indicating they have done such and verify the student is a US Citizen.  These records must be kept for a minimum of 5 years.  If the CFI cannot validate the student as a citizen, be done, they must submit the student’s information and immigration documents to the TSA and with for their approval prior to beginning flight training.

https://www.aopa.org/advocacy/pilots/alien-flight-training-program/us-citizens-seeking-flight-training