I am aware of this line in the Kodaly version too, Animaterra.
I looked around & found out that Fad could be derived from "fidfad" & asked an elderly friend if she knew what that meant. She laughed & said it's when somebody or somethin' gets all riled up," or when someone is "rather particular." This goes along with some of the info in the various dictionaries.
Then, I was just talking to my Turkish nephew who asked me what I was doing. Soooo--he says that he is aware of a term from his part of the world with the root sounding like "fad" which refers to something which sort of pops up or jumps up, & he told me it's sometimes used in an "impolite manner." He was just a bit embarrassed to tell me this! He did a search on the word, "fad" & came up with the term he was referring to as a possible origin for "fad."
I could be wrong, but--having seen many commonly sung oldtime lyrics & knowing that they were rather naughty, I'm betting that this naughty may be it & get some folks "riled up."