You can express the solution explicitly even if a != b as c = (a^n + b^n)^(1/n), but the issue is Fermat's theorem says the solution c has to be an integer (along with a and b). E.g. in the case a=b=1 we have 1^n +1^n = 2 = c^n, but c cannot be an integer for this to hold (with any n > 1). E.g. for n=2 you get c = sqrt(2), which is not an integer.

I doubt anyone is saying that there are infinite whole number solutions to a^n + b^n = c^n, as proving that would imply mathematics is contradictory.

I don't really know much about physics, but in math we don't use measure units that often. You usually define one base unit (like meters) and express everything else in it. Then you can just focus on the numbers and pure mathematics, which in all formality basically reduces to algorithmically pattern-matching strings.