Can the president of two different countries marry?

They can get married because many monarchs (sultan etc.) have been married in history. If the interests are incompatible and countries have democracy, this issue will not be very important. Someone either resigns or is dismissed. If this occurs in countries without democracy, then it is a separate debate whether the interest of the country already exists in the sense we understand today.

Note: The President and Monark are both spouses because they are the country's protocol chief.