In the early 19th century, a "bogus" was a machine used to make counterfeit coins. No one knows for sure how this coin-copying contraption got its name, but before long bogus became a name for funny money or for a fraudulent imitation of any kind. The more common "phony" adjective followed.
Mirage comes from the French verb mirer ("to look at"), which is related to mirror. Mirer, itself, is from Latin mīrārī ("to wonder at"), the ancestor of the commonly seen admire, miracle, and marvel.