Is it permissible for a father to give his poor, unmarried, 23-year-old son from his zakat money to help him start a project, knowing that he already spends on him?

It is not permissible for a father to give his Zakat to his son to start a business project, as long as the son is provided for by his father's sustenance; because in doing so, he would not be poor or deserving of Zakat. Zakat has specific recipients.