The geometric mean of n numbers is the n-th root of the product of all numbers. The mean square error is the sum of the squares of all numbers, divided by n. (I.e. the arithmetic mean of the squares.) They're not the same.
I'm not gonna edit what I wrote but you are interpreting it too way too literally. I was not describing the implementation of anything, I was just giving a link that explains why thinking about things in terms of area (geometry) is popular in stats. Its a bit like the epiphany that histograms don't need to be bars of equal width.