Is it permissible to count the money given to a man who claimed he lost his passport and needed money to return to his country as Zakat, knowing that I intended it as such when I gave him the money?

If you intend the money paid to be zakat (definition) and you predominantly believe the man is eligible for zakat because he is a wayfarer (ibn al-sabil), then that suffices for you; based on the Almighty’s saying: "Indeed, [prescribed] charitable donations are only to be given to the poor, the needy, those who work to collect them, those whose hearts are to be reconciled, to free captives, to those in debt, for the cause of Allah, and to the wayfarer; [this is] an obligation from Allah. And Allah is All-Knowing, All-Wise." However, if you predominantly believe he is not eligible, then it does not suffice for you.