There is a lot less urgency needed to setup a system for administering discounts like GitHub did for this reason.
They could probably just benefit from making it more clear that private repositories are free for students (even though they're free for everybody), no verification needed. If they reword it like that, it's basically saying "we're not even going to make you verify that you're a student and you get a great deal."
Also, my university gave me an alumni email address that I can still use today that has the same domain name.
Github lets me use my alumni account for the discount (I dont), their idea is if you want to abuse your alumni account you get 2 years free, and they just swallow that cost
EDIT: University/Colleges only. Per wikipedia: "Since 2001, new registrants to the domain have been required to be United States-affiliated institutions of higher education;..."
Many places that offer free tiers for 'education' only worry about that level of education and locale, anyway. (Not saying they should limit it to that, but many do.)
edu-domains where for a while also given to non-US institutions, but this stopped and has now been US-only for a long time.