Is a female student obliged to re-pay zakat on her money, given that her father used to pay it on her behalf annually and she does not recall having intended its payment, and is zakat due immediately upon receiving the money or before that?

You are not required to pay zakat (definition) a second time. Your father's intention, as your guardian, in paying zakat from your wealth is sufficient. It is not a condition that you intended to do so in your childhood or that he asked you for a power of attorney, for the Prophet, peace and blessings be upon him, said: "Indeed, actions are by intentions." The father's guardianship does not end merely with puberty, but rather with puberty, maturity, and proper handling of wealth, for Allah Almighty says: "And test the orphans until they reach the age of marriage. If you perceive in them sound judgment, then hand over their properties to them."