I am puzzled can Men With No King take control of dual-House character?

Response: After a Reinforcements event card is played,pay 2 gold to choose a character or location that does not have its owner's House affiliation. Take control of that card. (Limit 1 Limited Response per round.)

Depends on the affiliations. If one of them matches the owner's House card you can't take control of it.

A good parallel here is a character that has more than 1 trait. Let's say you have King Balon's Host in play, which has both the "Ironborn" and the "Army" trait. I play an effect that says "choose and kill a non-Army character." Can I choose and kill the Host based on its Ironborn trait, effectively ignoring the fact that it an Army? Of course not. Just because it is also an Ironborn character does not make it any less of an Army

Men with No King is effectively doing the same thing. Look at the way it is worded:

"...choose a character or location that does not have its owner's House affiliation..."

So, for example, if you are playing Greyjoy and an opponent plays Men with No King, anything that has the Greyjoy affiliation is not going to be an eligible target, even if it has some other affiliation in addition to Greyjoy . The fact that Theon or the Bandit of Winter also have the Stark affiliation does not make them have the Greyjoy affiliation any less. Just like the Host can never be considered "non-Army" just because it has the Ironborn trait, too, dual-House characters can never be considered "non-House #1" just because it has House #2, too.