You're making an argument from definitions that would exclude ML from being a functional language, which is absurd.
All functional languages need to bend the rules a bit in order to have side effects but in general the person is not writing functions and therefore is not getting the benefits of using functions.
FP is about reducing and restricting these functions that break the rules. Defining a class in terms of mutations and side effects is going against the philosophy of FP.
Which mathematical definition? A "box" is not a mathematical definition. There are multiple mathematically precise definitions of "function" possible. Yours is none of them. Some of them accommodate objects that model mutable state.
I don't think many (any?) PL theorists or functional programming practitioners would agree with your dogmatic stance. Where are you coming from with all this?