Bear in mind that moon and month have roots in Greek.
moon (n.)
O.E. mona, from P.Gmc. *mænon- (cf. O.S., O.H.G. mano, O.Fris. mona, O.N. mani, Du. maan, Ger. Mond, Goth. mena "moon"), from PIE *me(n)ses- "moon, month" (cf. Skt. masah "moon, month;" Avestan ma, Pers. mah, Arm. mis "month;" Gk. mene "moon," men "month;" L. mensis "month;" O.C.S. meseci, Lith. menesis "moon, month;" O.Ir. mi, Welsh mis, Bret. miz "month"), probably from base *me- "to measure," in ref. to the moon's phases as the measure of time. In Gk., Italic, Celtic, Armenian the cognate words now mean only "month." Gk. selene (Lesbian selanna) is from selas "light, brightness (of heavenly bodies)." Extended 1665 to satellites of other planets. To shoot the moon "leave without paying rent" is British slang from c.1823; card-playing sense perhaps infl. by gambler's shoot the works (1922) "go for broke" in shooting dice. The man in the moon is mentioned since early 14c.; he carries a bundle of thorn-twigs and is accompanied by a dog. Some Japanese, however, see a rice-cake-making rabbit in the moon.