Broadly speaking, there are three general paths that could lead to marriage between the young man and the princess.
The first is luck. That's not a minor thing. In fables and fairy tales, good guys have luck on their side. He might simply choose the right door by luck. (He might have chosen the right door no matter what she told him.) Or he might sneak past the tiger.
The second is cheating. The princess is wealthy. She could have set things up so there's an out, like having the girl show up at first, but then stepping in for her.
The third is love, which seems least likely. If the princess saw true love in the young man's face, and so did the lady behind the door, she might work with them to essentially, cheat again, on a more elevated level. Those involved might see the depth of love in everyone's face, and rework the scenario.