Since we use the term "moon" for the satellite of any planet should be have an official name for our moon?

We do have an official name for our moon.  Its official name is, literally, The Moon.

I know, it doesn't make sense.  We name all the other major celestial bodies in the Solar System after the gods the Romans thought they represented, which should make our planet's official name Terra, our moon's official name Luna, and our sun's official name Sol.

However, for some reason, when the IAU-- which is the governing body we've all decided has the authority to "make it all official" when it comes to astronomical matters-- got together and voted on what the official names of those three objects would be, they went with Earth, Moon, and Sun.

I don't actually call it the Moon.  I call it Cynthia, which is one of the other options.  There's also Artemis, Semele and Diana.  Luna is Latin for "Moon" but is actually a euphemism, the original Latin word being "mensis", which also meant "menstrual period", so it fell out of use.

Moon (big M) is the official name, in English, for Earth's natural satellite. It has official names in other languages, including Luna in Latin and Selene in Greek.

Used as a simile (moon with a small m), it is used for any object that behaves, in some way, like our Moon. The word "moon" used to mean anything in orbit around a planet (including "artificial moons" - like Sputnik) but it is now restricted to natural satellites that are thought to have formed in a natural process around their planet (as opposed to a captured satellite).

The word Moon comes from the same root as "month", which is the time period defined by the Moon going through its phases. 

Seven ... seventh

Moon ... month

same idea.

It is only in the USA that people were forced (by an astronomy magazine) to write the word moon with a small m, even when it refered to our Moon. The rest of the world has no problem accepting that Moon is the official name of the Moon, in English.

In the same manner, Sun (big S) is the official name of the star we orbit. We can use the simile "sun" (small s) to describe other stars that have planets around them (although we now tend to talk about exoplanets = planets outside our Solar system)

The Latin name for our Sun is Sol. From that, we get the adjective Solar (with or without a capital s) which refers to OUR Sun. 

The Solar system is the system of planets around our Sun.

Other stars can have planetery systems (or systems of planet).

Our Galaxy (big G) was long thought to be the only one. Hence it did not need another name. The Milky Way was the name of a band of light (made from the light of millions of stars located in the two nearest parts of the arms of our Galaxy. It is a much older name... and means roughly the same as the word "galaxy" (from the Greek word for "milk").

Thus, the name Milky Way (by itself) is meant for that feature. Calling our Galaxy "the Milky Way galaxy" is OK, since it is the galaxy that contains the Milky Way.

It is a 'moon' or a 'satellite'.  Other planets have moons, and they have names, so as to not confuse them with our moon.

The moon also has a name.  It's name is Luna or The Moon.  The sun's name is Sol or The Sun.

Whenever we read or hear the term "The Moon" we know that the reference is the the earth's satellite.  All the other known moons in our solar system have either proper names or numbers.  For us earthlings "The Moon" is a proper noun. Luna which is Latin for moon is a little mundane; Selene, a goddess of the moon doesn't seem to fit.  So let our moon be known as "The Moon". 

It's official the moon is luna

Look at all the planetary satellites and their names, ours just happens to be 'Luna' but we just call it the moon.

Earth. 1=Moon

Mars.  2=Phobos-Deimos

Etc,etc, etc....

Jupiter has 79 satellites

Saturn has 82

Uranus has 27

Neptune has 14

Pluto has 5

And they ALL have names or designations.

Besides this, there are 9 unrecognized or dwarf planets,

Ceres and Sedna the only with no satellites, and even those with,

all have names.

So you know, Luna was named LONG before any such thing as sci-fi.

Luna has a Capitol L

Our Moon also has a Capitol M

All the Planets, are given a Capitol to the begining of their names

 Of their Many Sattelites, they are known as moons with a lower case first letter

The reason is for to differentiate them from our own Sattelite

Although as their selves they are given Capitols 

The ancient Greeks called our Moon: "Diana"

too late now, it's in all our literature, our culture, etc.

It has to remain "The moon" or Our moon"