I did sign up for their student plan, I can confirm that they required me to give my student email address and take a screenshot of my student portal where it showed proof of my enrolment. Depending on whether your school is supported you may only need to provide your school email address.
I believe you only get access to the student prices temporarily (12 months) after that period you have to reverify your student status again.
Also another thing to add, Apple doesn’t do the verification it’s done by SheerID so it’s governed by their privacy policy, Apple doesn’t actually get any of your information they just know you’re eligible. (This is how it was for me in Australia could be different in other locations)