It's the lady. We are told that the princess loves the man. If she sends him to the tiger, that shows us that she does not love him, but instead loves herself, mistaking her love for herself for love for the man. But the omnicient narrator has established that she loves him. Love is, if nothing else, putting someone else before yourself. So if she actually loved him (and her love was not simply vanity) then she would endure the pain of losing him to her most hated rival in order to save him from her mad father's stupid arena.
The girls who laughed at the answer, "the lady," are obviously people who have never loved anyone. They assume the princess only loves the way the man loves her, and not the man himself, because they can't understand any other way to be. This isn't a question of the difference between men and women. It's a question of the difference between people who love and people who don't love. The people who love will assume that the princess will behave the way they would, and save the man, because they are told straight up that she loves him. The people who don't love will assume that the princess will be self-serving even at the cost of the life of the one she loves. If we hadn't been told that the princess loved him, there may be doubt. But we know that she loves him, so she saves him, because that's what it means to love someone.
Of course you save the one you love even if it means losing them! If love doesn't mean just that, then it doesn't mean anything at all.
Edit:
Here's another way of looking at it. Let's say it's not a man and a princess. Let's say it's a mother and a child. The child has to choose between a tiger and a new mother. The new mother is the old mother's most hated enemy, and the old mother knows that if the child chooses the new mother, the child will forget the old mother and come to see the new mother as the child's real mother. If you were the mother, which door would you pick?
Love is love. If the mother loves her child, she will protect the child no matter what hardship it means for herself. If the princess loves the man, she will do the same thing.