Is it permissible to consider the amount given as a loan to a needy person, which was intended as Zakat, as Zakat al-Mal or as fulfillment of a vow?

As long as you gave him the money with the intention of it being Zakat, and he is eligible for it, then this suffices you, unless that relative is someone whose financial support is obligatory upon you. And if you paid it as Zakat, it is not permissible to then consider it a vow.