Is it permissible to count the money that a husband gives to his family to cover the expenses of his siblings' studies in private universities as Zakat money?

Your husband is not permitted to count what he pays to his brothers as zakat, because zakat is not permissible for the wealthy or the strong, able-bodied earner. If they are unable to earn, then their maintenance is obligatory upon their solvent father. Similarly, university study expenses are not among the basic needs for which zakat is given, except under specific regulations and conditions. Your husband will be rewarded for assisting his father with his brothers' expenses.