IIRC the Greek/Roman goddess Artemis/Dione was called Selene when associated to the moon, just as her brother Apollo was called Febo (or Phebo or whatever is the correct spelling in English) when associated to the sun. I think Selene and Febo were older Latin gods that were later fusioned into Dione and Apollo when they adopted the Greek religion, but they were still called by those names when associated to the lights in the sky.
As it was mentioned earlier moon translates into my mother language, Spanish, as 'luna'.
So calling 'la luna' 'Luna' or 'el sol' 'Sol' sounds to me as redundant as calling the moon Moon.
Anyway I think that the correct statement is not "our moon is called Moon" (or Luna or translate into any other language you like), but "our natural satellite is called Moon".
More on Spanish:
'to land on the moon' translates as 'alunizar', as the normal verb for 'to land' is 'aterrizar' and derives from Tierra (Earth).
ISTR an article where it said that other words that derive from Tierra should be changed when applied to 'la Luna' such as 'luneo' instead of 'terreno' (piece of land).
And some recent news reports used the verb 'amarizar' when referring to landing on Mars.
ISTR some sci-fi story where the natives of the moon where called Selenites or some word derived from Selene. I guess that something derived from Luna would sound too similar to loony or lunatic.