The same goes for the Sun. Our sun is either proper name 'Sun', or if your prefer Luna above, Sol. Any other star with stuff orbiting around it is just another sun (noun).
I have also seen Selene or Selena being used from time to time, more often as a adjective (selenic) but also as a noun every now and then.
No you don't. Not in english at least. The moon or sun is understood in context and with the article "the". "The moon" and "the sun" without qualifications is understood to mean earth's moon or earth's sun. General references like "a sun" or "a moon" connote non-specific and non-identifying moon or sun. If you want to specifically identify other moons or other suns, you qualify it like - the moons of jupiter or sun of X solar system.
As an example, the recent SpaceX announcement.
"The proposed trip—a trip around the moon for two passengers lasting about a week—is almost the same as a scheme hatched by the tourism company Space Adventures, whose president told Quartz last year it had two clients booked for the mission."
https://qz.com/920993/who-are-the-space-tourists-elon-musk-w...
In Romance languages, there is a separate Latin term, closely related to the word "soil," which survives in French and Portuguese. Maybe sol/solo will be borrowed back into the Romance languages where it died out.
地 can also mean soil, but it’s a different word from 地球 meaning the planet earth. I don’t think it’s commonly used by itself to mean soil either, but I could be wrong.
Soil, or ground (on any planet) is a different word. There’s not much room for confusion.
That these are essentially generic nouns can show up more clearly in astronomical texts and of course in literature (science fiction).