From Max Weinreich's book, History of the Yiddish Language (translated from the Yiddish by Schlomo Noble, University of Chicago Press, 1973):
“Any Jewish community could partake of honey on the eve of the New Year to portend a sweet year. A similar custom could be found also among non-Jews. But the consumption on Rosh Hashanah of carrots (mern) in allusion to the verb zikh mern (to increase) and its combination with a prayer to the effect that “may our merits increase”—this could only be an invention of Ashkenazic Jews. Kol mevaser (a voice proclaiming) is recited by Jews on Hoshana Rabbah everywhere, but only speakers of western Yiddish have the custom of eating cabbage soup on that day in allusion to the German Kohl mit Wasser (cabbage and water). In a township near Rovna, Volhynia, water carriers used to celebrate on the Sabbath of the weekly lection of Emor (speak) in allusion to the emer (bucket, in Yiddish).” (p.5)
This makes me wonder if the creative and clever traditions that arise out of homophones (or near-homophones) are more effective or appealing when they have to do with food. Also, I think it's wonderful that Yiddish pronunciation is just far enough from the German that mehren and Möhren become the same, and that mevaser is just close enough that someone probably chuckled and said, "wouldn't it be funny if..." and then it became a tradition.