Does the flash drive your brother found become your property, especially since neither of you has found its owner or any information about them?

If the address of the person who put the memory card there cannot be known, then it is considered like a lost-and-found item (luqatah), and its rulings apply to it. Thus, it must be publicized. If its owner comes and is identified, it should be handed over to him. If he does not come, it is not permissible to use it until a year passes. After a year has passed, it is permissible to benefit from it, while guaranteeing it for its owner should he appear, based on the hadith of Zayd ibn Khalid: "Know its container and its tie, then publicize it. If its owner comes, (hand it over to him), otherwise, it is up to you."