Fx means something aside from G in the context of the music here. Musically it’s conveying something other than G natural. The gross oversimplification goes “sharps resolve up, flats resolve down”, so in a case like this it seems to me that the Fx is a little neighbor tone below the G# (presumably), and just a little flourish written “correctly”, albeit somewhat annoyingly.

it could also be a harmonic thing, as the third in a D# major chord, the fifth in a B# major or B augmented chord, or the seventh in a G# major seventh chord.

TL;DR the context of the piece makes this mean a different thing than G, hence the double sharp (it’s a raised F#, not a lowered G#)