Fortunately, Wikipedia exists to solve such problems.
Apparently, maccherone means "boorish fool" in Italian and young British men who spent time in Italy tended to come back describing anything very fashionable as being "very macaroni." It evolved to include anyone who dressed or even spoke in "an outlandishly affected and epicene manner."
So, the joke of the song (which was most likely written by the British to mock Americans) was that this American would-be fop was naive enough to think that putting a feather in his hat was all it took to qualify as fashionable. Americans, no doubt craving songs about themselves, happilly adopted the song as their own, either unconcerned or unaware that they were being made fun of for their distinct lack of style