Yes, they are. This case is not about that, as it is not in dispute.
> They are paying to be in school.
Indeed, they are. And, as such, the are, to that extent, customers of the school, and legally treated as such.
> They may be paying for it with money gained from the University but they are still students.
Here's where the issue is. Yes, they are still students. But, when they are paid by the university for work which benefits the university (such as work as teaching and research assistants), they are also employees.
> Should an undergrad that is on a university scholarship be considered an employee too?
An undergrad on a university (academic, sports are potentially a different issue and the subject of some active controversy right now) scholarship is generally not doing work that benefits the university. OTOH, an undergraduate paid by the university as a tutor (undergrads generally aren't employed as TAs) or research assistant (a role in which undergrads are not uncommonly employed on campus) is, uncontroversially, an employee.
> How are the economic realities of grad school any different from undergrad?
The only difference is that graduate school programs often bundle paid work of the type that would, uncontroversially, make anyone else an employee as part of the requirements of the program. This ruling simply says that bundling a requirement to do paid work on behalf of the university into the requirements of an academic program doesn't magically transform that work into a non-employment relationship.