What should one do whose bag was replaced with another at the airport by mistake, and its original owner did not report it? Is he permitted to make use of the bag he now possesses?

The bag in your hand is a found item (luqatah), and you must publicize it by registering its details at the airport where you took it, as its owner is searching for it. If its owner does not appear after one year, you may benefit from it, while guaranteeing it to him whenever he is found. As for your lost bag, you have the right to demand its return from the concerned authorities.