One of them, I assume the Girl, fell in love with the other as they fought (A sort of one-sided enemies to lovers). When this person manages to get enough feathers, instead of wishing for whatever their organization wanted, they wish that the other "married them" (wording may vary, but that's the general idea). Because of how the wish works, it sort of "implants" the idea of the two falling in love to the other, and thus kicks of the story.
This would explain a lot of things:
1. Why there was no discriptions of how they feel in love. It's just vague ideas and statements that tells us nothing. (e.g. how the male MC said that "after many twists and turns, they started going out".)
2. Why we don't know how the organizations disbanded. The feathers were used. They no longer have any reason to exist.
3. Why the wife doesn't want to sleep with him. If you assume that she was unaffected by the wish, then she's practically just started daiting. She
is shy, as she has not gotten used to the idea of being intimate with his, as she otherwise would have over the daiting period. One would think that they would have, at the
very least, discussed it before getting engaged of married.
4. Why he's so madly in love with her. That was her wish.
It's sort of like that one webtoon,
How I Failed To Save The World.
One last thing: The guy being shown at the end of chapter 4 feels like a bait and switch: He's going to be chill, not an antagonist.