No, you can't say you have customer service experience. You are comparing apples to oranges since those are your pupils and not your customers and can't be compared in terms of authority-relation, communication-style, behaviour etc. and hence you should not mention to have customer service experience.
Just imagine speaking to or treating a customer like your ...
Does teaching experience count as customer service experience?
No. At least not in the context of the online job application.
During an interview, you could explain why you feel they are similar and how experience in one would contribute to success in the other.
Good answers already, my addition is:
but it is actually true in a certain way from my experience
Sure it is, but not in the way they mean it. So listing it as such makes you look out of touch or desperate to put anything down.