We're gonna have to make the Moon's name more specific if we ever survive long enough to colonize other planets. It's like if Earth's name were Planet.
I mean, yeah, many celestial bodies take their names from Latin. Like how the proper name of the sun is Sol, which matches solar, the lunar object would be called Luna. Selene feels like a retronym to match other Greek deities, and would be like renaming Earth to Gaia
either of those options is a fair pick given their oldness
Luna would definitely rank over Selene in the Western astronomical tradition. As you say, its name is just The Moon, which is what Luna means in many current day Latin derived languages
I'd be totally up with renaming out Tierra and Luna to something that is not eurocentric. Would be a nice change of pace against how much of immediate astronomy is caught up in remixes of Greek and Latin.
"Earth" literally just means "the dirt under your feet".
But most of us don't identify our location so broadly. We say what state or country we're in, and identify culturally that way. I imagine that won't change when there are people living on the Moon. They'll identify by the name of the base or settlement they live in.