The Moon is a unique satellite of the Earth.
is grammatically correct, however, it is ambiguous and may not mean what you think it does.
If what you meant to say is that Earth only has a singular satellite then the way to say it is :
The Moon is the only satellite of the Earth.
If what you meant was that of all of Earth's satellites, the Moon is the only one of its kind, then
The Moon is a unique satellite of the Earth because we never see its far side.
Unique is a funny word and well, it's, unique (bad pun intended...)
One trick I use to figure out whether to use a or the is to look at the phrase in the absence of unique
1) The Moon is a satellite of the Earth.
2) The Moon is the satellite of the Earth.
3) The Moon is a unique satellite of the Earth.
#1 sounds good while #2 doesn't, therefore use a in #3