If you took an apprenticeship, you wouldn't think you're a customer. If you took an internship, even if you had to pay for it, you wouldn't be a customer. Students are customers. They're students.
The model that seems to underlay this idea is that in a financial transaction, one person is a customer, and other is the provider (of a service, or product, etc.). But not all financial transactions fit this mental model of what is a kind of social relationship.
Suppose you go to Craigslist, and find that someone is selling their old bike. You give them a call, give them cash, and you get a bike. Were you their customer? Something doesn't fit. This is a one-off transaction, on each side. There is no "custom". But I don't want to argue the point, because there are other examples.
Suppose you get the opportunity to pay for an internship at a prestigious lawfirm. You pay your $1000 bucks, and then you work for free for a few weeks. Are you their customer? Sure, there was a financial transaction, but the relationship isn't one of being a customer. Is it?
Ok, suppose you are a plumber. Every few years you pay a fee to a certification board to keep your license. You also pay membership money to the Better Business Bureau. Are you customers to the certification board, or to the BBB? Doesn't seem like it to me.
Students are students. That is the relationship they hold to schools. That there is also a financial transaction happening is incidental to the primary relationship. Indeed, for students who pay no money because everything is funded by grants, then the idea that students are customers breaks down further.
I understand the temptation. It is a lot of money most people pay for an education. But students are not customers, and university degrees aren't products.