For those who didn't catch the reference, "the one in Odaiba" likely refers to the real sized Gundam that was build in Odaiba.
The fact that it has a mustache is also a reference to the Turn A Gundam, a Gundam that was found in ruins, and instead of an antena, it has a mustache.
In Gundam Lore, Turn A Gundam is the series that made all alternate universes being "canonical" with each other, with the idea that the Turn A Gundam would appear and "destroy the world" back to how it was. It's treated as the most powerful Gundam, although the version that appear in the anime is essentially a powered down version with not a proper pilot, but the original one from "Dark History" was a beast that could teleport, had a rifle akin to a Colony Laser (A laser the size of a space colony), and both the Gundam and the pilot were essentially invincible because it could be reconstructed via its Nanomachines. It's main attack, the Moonlight Butterfly, spreads Nanomachines everywhere and can reach Jupiter.
In short, it's a world ending Gundam, created as a copy from the Turn X, a mysterious machine that came from space, which is believed to have been created by humans that have spread through space aeons ago. The Humans from Earth found this machine, and created the Turn A based on it.