nevau, noting that the english words meaning moon, month are cognates, wondered to what extent the same is true in other languages with different lineages, especially the non-Indo-European languages.

An interesting question. I took it to the KudoZ site, where translators, stumped over a word or phrase, can get help can post questions and get help from their peers. The site covers dozenns of languages from all over the world. It's not geared to such general, multi-language questions, but was very helpful. Here, consolidated, are the answers received:

wordcrafter: If I understand correctly, the words for "moon" and "month" are identical or are cognates [in] German, Sanskrit, Irish, Lithuanian, Avestan, Persian, Old Irish, Welsh. In other languages (Greek, Armenian) they were originally cognates, until another term was substituted for "moon".
Jerry Thomas: In Mandarin Chinese the word for month is the same as the word for moon.
John Pilgrim: German Mond/Monat;Dutch Maan/Maand. The Hebrew word for month, khodesh, although not similar to the various words for moon, is etymologically related to the word for new, khadash, as in a new moon. See here. [follow-up] Aha! In both Quéchua and Aymara, month and moon are the same word: Quéchua: killa. Aymara: phaxsi. (see here and here).

willmatter: Strangely enough, the Romance languages have a weird consistent shift, all of the "moon" words are "lun*" (lun +something), examples include "luna" (Spanish), "lune" (French), "lua" (Portuguese) and "luna" (Latin) but all the "month" words start with "M" and strongly resemble each other. For example, Spanish is "mes" and French is "mois". For Japanese, the actual noun for "moon" is "tsuki" but for compound words, phrases and kanji combinations "gatsu, getsu" come into play. Also, you combine a counter with "getsu" to count months of time. Hebrew: Levanah/Khadesh (sometimes spelt "Chodesh"). Yiddish: Levoneh/Khoidesh.
Languages where the word for "moon" and "month" are identical: Croatian "Mjesec"; Frisian "Moanne", Indonesian "Bulan"; Inupiat[/b] (Eskimo) "Tatqiq" (the "q" is a hard "k" sound in the back of the throat), Korean "Tal"; Romanian "Luna" (with an accent mark over the "a"); Serbian "Mesec"; Swahili "Mwezi"; Tibetan "Da-wa"; and Turkish "Ay".
Next, here are a few with some minor differences, the word for "moon" comes first, followed by the word for "month": Danish, Swedish and Norwegian have "Måne" and "Måned"; Afrikaans "Maan" and "Maand" (not surprising considering its relationship to Dutch); Esperanto "Luno" and "Monato"; Finnish "Kuu" and "Kuukausi"; Cheyenne "Taa' eese 'he" and "Ese 'he".
Elena Sgarbossa: Another ramification in Spanish aside from "luna" (moon), is "lunes" (Monday). Italian: luna & lunedí (and month= mese).
Kirill Semenov: Russian "month" sounds as "mEsiats", and we have two words for "moon" -- the same "mEsiats" and "lunA".
Nesrin: In Farsi (Persian), the word for both moon and month is "mah". and "maah" in Urdu. And, as is often the case, it probably all began with Sanskrit: Maas in Sanskrit also means moon and month.
Dan Mencher: Czech 'měsíc' means both 'moon' and 'month'. Hawaiian'mahina' and 'malama' both mean 'moon' and both mean 'month'. Interestingly, the two words are separate in Gaelic: 'moon' is 'gealach', and 'month' is 'mìos'. In Estonian both 'month' and 'moon' are 'kuu'. In the Alabama Native-American language 'month' is 'hasi' and 'moon' is 'tankahasi' (the connection can be seen in the ending of the word for 'moon'). In Cree there are two words for 'month': 'pisim' and 'wok'. There are also two words for 'moon': There are also two words for 'moon': 'tibiskáwipisim' and 'wok'. Tlingit (spoken in parts of Alaska), the word 'dís' means both 'month' and 'moon'. In Wolof (spoken in West Africa), both 'month' and 'moon' are 'weer'. In Hausa (spoken in Africa), both 'month' and 'moon' are 'watàà'.

I can add a couple more from unrelated families: Chechen butt 'moon, month'; Georgian tve 'moon', mtvari 'month'. I don't know enough about Georgian morphology to tell if mtvari could be related to tve.

The 'strangely enough' comment about lun- vs mens- is strange. There isn't a consonant [or consistent] shift, because they're different roots. Latin luna comes from Old Latin *louksna, from a root 'shine, be white'. And of course the words in Spanish, French, Italian etc. resemble one another: they're closely related languages!

As well as Spanish and English, Monday is also some form of Moon-day in French (lundi) - and presumably the other Latin-based languages - and Welsh (dydd llun).

What interests me here, as well as the linguistic similarities, are the cultural ones - the fact that so many societies have chosen the same symbolism for naming the days of the week. If the root of the word is a different god, (e.g. dydd Gwener - Venus, and Friday - Freya), the two gods are each others' (ap?) equivalent, more or less.

It just shows how we're all related! Now, if only we can get the warmongering leaders of so many man-made countries to work that one out...