The Unfinished Vault, also known in-game as Fake Vault 13, is merely a small cave. There are some construction signs lying around indicating it was meant to be a Vault-Tec Vault, but it was not finished before the Great War. Its intended number is unknown.
It must be noted that the cave for the fake Vault 13 looks exactly like the cave for the real Vault 13.
One of the caves used during random encounters has the exact same layout, apart from the construction signs.
The Unfinished Vault appears only in Fallout 2.