In the US, you start as a not for profit and

unless you are a total fool, you quickly add an appropriate non-profit status appropriate for your activities [501(c)3 is only one of about 14 IIRC, albeit the most advantaged] because that's the one that gets you out of paying federal taxes. NPOs are not necessarily prohibited from making money, it's just that their directors can't profit from it. Usually the monies go back into the community is some form. If you don't get the NPO status, you have to either balance your books damned carefully, or pay 35% direct to Uncle Sam. So I'd bet they have an NPO status of some sort, although not necessarily the educational one.

IANAL but I've helped organize and run many NPOs in my time. Even worked for one once upon a time.