(For those not aware: the song is from ניקוי ראש Nikui Rosh, lit. ‘head cleaning’, Israel’s second ever satire show on TV. It was extremely influential, to the point people avoided planning events like weddings and bar mitzva celebrations when it was airing lest people didn’t come. Some of the actors on the show later became celebrated veteran actors. The song parodied in the middle is this jingle, advertising a brand of soaps, whose tune became so iconic every future ad for that brand has used a variation or a snippet thereof.)
Well, these two homophones are not related. As is shown by Aramaic תּוּמָא garlic, this word had a proto-semitic [θ] like in thief, but שׁוּם is a secondary form of שֵׁם name, whose [ʃ] like in ship is original. By the way, בִּצָּה [bitsa] swamp and בֵּיצָה [be(y)tsa] egg sound different.