Why is it rare to find the phrase "a saggy mustache" rather than "a droopy mustache"? I googled the phrase "a saggy mustache" but it showed very few results, compared to "a droopy/drooping mustache" which is so common that it is included in dictionaries' examples. This is while "droop" and "sag" are synonymous and both mean "to hang down or bend".
I agree with Sven Yargs - a sagging moustache could be falling down in the centre (maybe.. It's a fake moustache...?) whereas droopy is hanging down at the sides. Sag implies something that once was straight but now arches downwards - a line that has gone 'off-true'. Saggy moustache would be poetic, it's not really a normal expression, whereas droopy moustache is.
Droopy typically implies that the edges hang down while the centre is supported. So imagine pointing your index finger straight up into the sky, and you place a tissue on it, the centre of the tissue will sit on your finger, while the four corners will droop down.
Saggy implies the the centre lacks support, while the edges have support. So if you were to hold the four corners of a tissue, the centre would "sag" down due to gravity.
So by looking at the shape of a moustache, which looks like "∩". The edges point down, therefore moustaches are droopy and not saggy.