The Sea Dragon God enchanted her with the memories of the very first shrine maiden who was the dragon's daughter. He explains the story of how this entire "tradition" and ritual, as well as the blessing on the city, was made in the first place. The dragon princess fell in love with a human and chose to give away all her powers in exchange for having a humanoid form. The people of that city thought that she was a monster disguising herself as a person in order to do evil things to them, and thus they attacked and rejected her. The only one who stayed by her side was the human guy who later got called the Sea Dragon Brave/Hero. In his anger, the Sea Dragon God almost destroyed the humans and the city with them. But the dragon princess and the hero stopped him, and he gave them his blessing. This blessing affects the entire city now because the dragon princess and the hero were both selfless and wished for everybody to be happy, not just them.

What the Sea Dragon God now wants Nagi to prove is whether or not he actually cares for Iris/the Dragon God's child, as what the two of them are requesting is for Iris to be freed from the responsibility of having to do the ritual that keeps the city blessed. When that happens, Iris will likely lose her standing of being the child/descendant of the Dragon God, and yet the blessing will still keep on being given to the city--essentially making it a freebie for the descendants of the people who hurt the Dragon Princess in the past. A lot of people have already shown dislike of Iris because they think she is a part monster (she has scales). So without her position of power as a shrine maiden she becomes vulnerable. The Dragon God wants to be assured that his descendant will have somebody to protect and care for her if that happens.