"A dual nationality" vs. "dual nationalities"
Solution 1:
My vote is for "He has dual nationality." This construction treats "dual nationality" as a legal status unto itself, not as countable item(s).
The Google test agrees with this, returning 9,000, 68,000, and 3,000 results for your three respective options.
Solution 2:
I would say it was "dual nationality", cause although he has two types of nationality, he has one actual nationality, that is a mix of the two.
This can also be seen in things like "the dual number.", which is a number composed of two other numbers.