How did you arrive at the "correct" answer, by the way? Your original answer is that given by my mark scheme and is the one redox equations seem to come to, but it is true there are infinite solutions to this problem. How then did you go about finding the correct answer?
You can get the correct answer assuming oxygen is produced only from the hydrogen peroxide, and not from the permanganate. In other words - oxygen atoms are not equivalent here.
While this approach gives the correct answer here (read: it reflects the stoichiometry observed in glass), I am not convinced it is a correct approach in general. Don't ask further questions - I simply don't know.
And is it then safe to say that if there isn't 1 less equations than there are variables, there is no unique solution?
Even having n equations is not guaranteed to give solution, as they have to be independent.
Generally speaking when you have n unknowns and less than n independent equations, there is no unique solution (this is math, not chemistry, such system is called underdetermined). In the case of balancing we have additional condition that all unknowns should be integers and the smallest possible ones, which is in a way equivalent to adding n
th equation.