In "The Lady or the Tiger?" what personality traits does the king's daughter have in common with the king?

The king and his daughter are both semi-barbaric.  This means that they are passionate and strong-willed, and make sure that things are done their way.  The king does what he wants and doesn’t ask for input from anyone.

He was a man of exuberant fancy, and, withal, of an authority so irresistible that, at his will, he turned his varied fancies into facts. He was greatly given to self-communing, and, when he and himself agreed...

The king and his daughter are both semi-barbaric.  This means that they are passionate and strong-willed, and make sure that things are done their way.  The king does what he wants and doesn’t ask for input from anyone.

He was a man of exuberant fancy, and, withal, of an authority so irresistible that, at his will, he turned his varied fancies into facts. He was greatly given to self-communing, and, when he and himself agreed upon anything, the thing was done. 

The king develops a sort of wacky system of justice.  He builds a huge area that is part courtroom and part entertainment venue.  The people come to want the accused choose between two doors.  One door has a tiger behind it that will maul the accused man to death, while the audience watches.  The other has a beautiful lady what will marry the man, while the audience looks on.  Either way, they get a show.

The king’s daughter has “a soul as fervent and imperious as his own” and is the apple of his eye.  He knows that she is very passionate and semi-barbaric too, so of course she would take a lover.  When he finds out about one and he doesn’t approve, he throws him to his special system of justice.

The daughter, like the king, wants things her own way. She finds out what is behind what doors, even though it’s a big secret.  Her lover is aware of this, and looks to her for a signal.

He had expected her to know it. He understood her nature, and his soul was assured that she would never rest until she had made plain to herself this thing, hidden to all other lookers-on, even to the king.

We do not know which door she led him to.  She was definitely the jealous type.  She did not like the woman who was chosen to go behind that door.  Even if she let him live, she would have to watch him with that woman. A semi-barbaric princess might be more likely to say that if she can't have him, no one can.