Is my wife's brother entitled to recover the money he paid for my car after he backed out of the purchase, and is it permissible for me, according to Sharia, not to return the money to him?

Your wife's brother does not have the right to annul the sale that took place between you. A sale, once it fulfills its conditions and pillars, becomes binding on both parties. It is recommended for a Muslim to grant annulment to his brother, for the Messenger of Allah, peace and blessings be upon him, said: "Whoever grants annulment to a Muslim, Allah will annul his slip (difficulty)." Annulment (iqalah) is the cancellation of a contract between two contracting parties, and it is legislated by scholarly consensus. It means that whoever agrees to annul a sale for a Muslim who regrets his purchase, Allah will remove his hardship and difficulty on the Day of Resurrection, as a reward for his kindness.