What are the consequences for someone who swore to fast one day for each specific sin committed, but then forgot the wording of the oath and the number of days, and committed this sin many times, while being a university student financially dependent on his father?

A Muslim should not make frequent oaths. If one is unsure about the repetition of an expiation, then only one expiation is required, because the default state is the absolution of one's liability. Similarly, the default for a conditional vow is the minimum that the name applies to, and fasting one day is sufficient for you. It is permissible for you to pay the expiation from what your father gives you; otherwise, you must fast three days. You must also sincerely repent from the sin.