My understanding is that it comes from the French
étoupe which means: fibrous component produced during the scutching and / or combing of bast textile fibers, in particular flax and hemp (I am translating form a French dictionary). The one translation I have found in English is “cotton waste”. The pic above is this, not a toupee.
I didn’t know taupe was an English word? In French
taupe literally means ’mole’. But
étoupe is something else.
I have just asked my French bf to confirm the above as French is not my first language!