The barbaric nature for the king and his daughter found in their sense of justice and entertainment. The king didn't care about discovering how was actually guilty and innocent, he wanted to leave everything to chance and allow the justice system to provide entertainment for himself and others. He is also barbaric in the sense that his reward system for being innocent completely disregarded the feelings and desires of the person on trial. He is not barbaric in the sense that he does want to have a justice system and to be fair in his rulings, but the system itself reflects his semi-barbaric nature.
His daughter has the same traits. She is truly conflicted about how to deal with the trial of her love. She wants him to live, but she doesn't want to see him married to anyone else. She takes charge of the situation by discovering which door contains which fate, but her how she directs him remains a mystery.