Is it permissible for me to take profit from a car I did not buy or pay for, merely because my cousin gave me a portion of his profit from it?

If you did not buy the car, then the profit belongs to your cousin. However, if you bought it and he sold it without your permission, then the transaction is unauthorized, and the sale becomes valid if you ratify it, and the profit will be yours. His statement, "It is yours," is not sufficient for the sale to be concluded unless the conditions for a valid sale are met, such as offer and acceptance, and knowledge of the price and the item being sold. Whoever buys an item is prohibited from selling it until he moves it from the place of its first seller, based on the saying of the Prophet, peace be upon him, to Hakim ibn Hizam: "If you buy an item, do not sell it until you take possession of it," and his saying to Zayd ibn Thabit: "He forbade that goods be sold where they are bought until merchants take possession of them and move them to their own places." This includes all goods, and the "possession" of movable items is usually by moving them to a place where the seller has no claim. There is no blame upon you for accepting money from your cousin if it is to appease you.