I am wondering which is the proper way to say that one thing or the other is required.
Only one of passport and national id is required.
Only one of passport or national id is required.
Only one of is effectively saying Only one of the following list. It is followed by a list of two or more items, and therefore requires and.
In your second version, with or, the one of is superfluous, because the or implies a choice of only one item, as in "Choose A or B or C ..." (or more usually "Choose A, B or C").
Either form can be used with two or more choices, but if there are only two choices then the or form (without one of) is briefer and clearer (in my humble opinion).
If the statement were that you need two of X, Y and Z, then using or would be wrong, because (1) or implies a single choice, and (2) with picking two or more you are choosing from a list, and a list requires and (not or).
The either/or construction lets the reader know that only one of the two is required.