Is what I give to my father and brother considered zakat for my money saved in the bank, or must a separate zakat be given for the saved amount despite financial burdens?

It is not permissible to count the expenditure on a father from Zakat (definition), due to the obligation of his maintenance upon you. As for spending on a poor and needy brother, it is permissible to count it from Zakat if it was intended as such when disbursed. So, if you give your brother what fulfills his need with the intention of Zakat, nothing further is required of you. However, if it is less than the obligatory Zakat, you must pay the remainder if the wealth reaches the nisab (definition) and a year has passed over it.