Our moon is officially called Moon. Luna is just something used occasionally, but not always. The sun is also officially named Sun. It's just that those terms started being overloaded after we learned that there were other Moons and Suns. You can try to get all fancy with Sol and Luna, but those are just old Latin terms for Moon and Sun. They're no less generic than the ones we currently use.

https://en.wikipedia.org/wiki/Moon#Names_and_etymology
https://en.wikipedia.org/wiki/Sun#Etymology

The official name is "The Moon", Luna just means moon in Latin. It's all about capitalization. Moon is our moon, and moon is any moon.

Luna is the Latin word for moon. It’s not the official name in English. It’s just called “Moon” in English, just like the sun is just called “Sun” despite being many other words in other languages.

Because there is just one of them it’s not really necessary to give it a unique name, but there are lots of options in Latin, Greek and other languages if you want to get fancy.

Except "luna" is latin for "Moon", which means that in many neolatin languages the problem is still there. Probably Selene would be good for every language, She's the whatever goddess of the whateverness of our Moon or whatever

My cat is called Luna 🐱

That fucker has followed me around the world peaking into many places I've been. I should at least know who's stalking me.

That it is always around. Just hanging there in the sky. That we've spent so many nights together. But I guess it is in a way a long-distance relationship. Also its just a pale reflection of the sun. A rocky relationship, in many ways.

When I see people on the trail I sometimes only greet the dog.

What like Kleenex?

Moony McMoonface

Terra?

Possibly. Better be more poignant than "Alpha".

I think every celestial body we know of already has a unique name/ID