We seem to have named every moon orbiting other planets. Why haven't we named our own moon? And for that matter, why doesn't our sun have a name since we name or number stars?
"The Sun" is fine as long as you're not leaving our solar system; less so "the Moon" when there are hundreds of planetary satellites in this system alone.
In science fiction as varied as Isaac Asimov and ''Star Trek'', the names are Sol and Luna.
EDIT: My point is that sci-fi authors are writing from the point of view of societies with many "suns" and "moons", and those societies have adopted the classical terms as the "current" official names.
What is a name?
A name is a word, that is reasonably unique, that is used to identify a person or thing. When a child is born there is no word that identifies it, and so its parents have to choose a "name". Similarly when a new astronomical object, such as an asteroid, is discovered, it has no word that identifies it and so it is assigned first a number and later a name.
But there is already a word that identifies our sun and moon: "the Sun" and "the Moon". There is no need for a new name because they already have a name. Notice that English (unlike some other languages) gives us a typographical clue that these are names, because the first letter is capitalised. The Moon is a moon of the Earth and the Earth is a planet orbiting the Sun.
Earth, Moon and Sun are the correct names in English.
Of course the Sun and the Moon have names. The names of those objects in English are "the Sun" and "the Moon". Note the use of a definite article ("the") and the capitalization of the names themselves to indicate a proper name. The Sun and the Moon have many names, at least one name for each for almost every language ever spoken. The translations of those myriad names into English are inevitably "the Sun" and "the Moon". Being the two most obvious celestial objects, those names have been around for a long, long time, almost certainly predating writing.
On the other hand, the knowledge that the Earth is a planet that orbits the Sun is rather new, less than 500 years old. The knowledge that other planets have moons is newer yet, about 400 years old. The knowledge that the Sun is a star is even newer, less than 200 years old. Giordano Bruno was burnt at the stake in 1600 for suggesting that the Sun is a star. It wasn't until the 1830s when scientists were able to measure the enormous distance to even the closest stars.
It is named 'Sol' and the moon was named 'Luna'. Hence the term Lunar explorer and 'Solar' System..
Other cultures have used different names for our Sun and Moon (for thousands of years) - So actually they have multiple names each.
Most people refer to them simply as "The Sun" & "The Moon".
Well, even though one may encounter some "poetic" names such as Sol or Luna, I believe these bodies are referred to as "the Sun" and "the Moon", and that is the only formal name I've heard of.
As for the reason, I'd say it's because there is no need for special names like Sol because they aren't really used in practice (we've known of the Sun and the Moon for quite a bit), while it is quite important to name newly discovered objects so we would know what to refer to them as.
I did not find any definite reference, but if the Sun and the Moon had any other official name, it would have come from the International Astronomical Union (IAU). The only reference I found was a mention in passing of Sun and Moon (capitalized). https://www.iau.org/public/themes/naming/ There may be more information hiding in some publications. But if they had another official name for these two celestial bodies, it would seem reasonable to expect that it would be featured fairly prominently on their Web site.
It would in fact make some sense to have distinctive names, to avoid statements like "Alpha Centauri is the sun of another planetary system" where sun does not refer to our Sun. That may have been the idea behind the notion that there is a distinction between "Sun" and "Sol". But it does not look like the IAU backs this up.
As best as I can tell, that idea may have originated in Hollywood, specifically Star Trek. Since I do not have any definite information to back that up, take that with a grain of salt.
Because there's no need to. "Sun" and "Moon" are relative terms and it's clear which bodies we mean (as long as we say the Sun / the Moon) when on Earth. If you stood on Proxima b's surface, Proxima Centauri would be your Sun, Proxima b's spherical satellite (if it has one) would be your Moon, and the Earth's Sun would be an average star in the night sky we'd have to find a name for in such case (e.g. Cassiopeia VI since it would enlarge the constellation of Cassiopeia by another star). But as long as you're on Earth, it doesn't matter.
You can even use the term "Earth" relative for another planet you're on in the sense of "land", just like we talk about a "geology of Mars" (geos is Greek for 'Earth').
There isn't a general naming scheme. However, we can leverage a pattern for naming the apsis of orbits. We historically named these on a per-host body basis, so apogee and perigee were the names for the furthest and closest an orbit gets to the Earth, respectively. The -gee suffix is derived from Gaia, so that would be the name for our planet. The solar apsis are apohelion and perihelion, the -helion suffix being derived from "Helios," the Greek god of the sun. Lunar apsis have used -lune, -cynthion and -selene, deriving from names Luna, Cynthia, and Selene.
These are, of course, just one of many names that could be used, but the pattern of using historical gods would have value in such a situation. We might talk of a "new Earth" or a "new Sun" to describe a planet or a star that is similar in characteristics to our planet or star, so those names might shift from nouns to adjectives, describing planets or stars. However, there is little reason for a historical deity to be associated with a celestial body that was never seen by that historical society. It is unlikely names like Luna or Helios would ever be applied to bodies outside our solar system. Whether they would get applied to bodies in our solar system is a question for the linguists. It would depend highly on how we adapt other words like "sun" and "sol" in a multi-solar-system society.
And that linguistic adaptive approach would answer your question as to why we don't have names for them. Currently "sun" is sufficiently descriptive that it acts like a noun in our speech and needs no additional context. "The moon," with the pronoun "the" included is sufficiently descriptive. For the ancient Greeks, there wasn't even a need to identify the Earth's moon from a moon of Jupiter, because they didn't know Jupiter had moons.
For all I know, we'll name "the Sun" "Sun_00001" to disambiguate. But there does seem to be a history of leveraging the ancient deities when disambiguation is needed in a language.