The Moon (月, Tsuki) is the home of the Lunarians, moon rabbits, and of Eirin Yagokoro, Kaguya Houraisan, and Reisen Udongein Inaba. In modern times the Lunarians and moon rabbits are fighting a war against the invading humans of Earth, but based on Silent Sinner in Blue this invasion was ultimately unsuccessful.
PMiSS lunarcapital

The Lunar Capital as depicted in Perfect Memento in Strict Sense

The path between the moon and the earth only exists during a full moon, which is why Eirin and Kaguya conspired to hide it during the events of Imperishable Night.

Staring at the true full moon is said to be able to induce insanity in humans.

In the distant past the people of the moon tried regulating people of the Earth by creating monsters, according to Kaguya in Imperishable Night. Also according to her things such as magic originated from the moon- "The moon gave many powers to beings of the earth. Many of them, such as magic, were originally the moon's power."
SSiB7 MoonPalace

Could this be the same building as the one in the foreground of the picture above?

More than a thousand years ago Yukari Yakumo launched a failed invasion of the moon after gathering a youkai army. Her exact reasons are currently unknown.

The Magical Astronomy CD liner notes depict tourism to the Moon being commonplace in the outside world, during the near-future time inhabited by Maribel Han and Renko Usami. How this affects relations between humans and Lunarians is currently unknown.