Is Zakat obligatory on the total annual salary at the beginning of the following year (the salary of the first year at the second, and the salary of the first and second years at the third, and so on) for saved money?

What you were told about the method of paying Zakat for past years is correct, except that you are not obligated to pay Zakat on any month's salary until a year has passed on it, and it reaches the nisab (minimum threshold) by itself or in combination with other wealth. However, it is permissible for you to pay it in advance, and that is better for the poor.