In what way is the princess similar to her father in "The Lady or the Tiger?"
The king and the princess’s father is described as “semi-barbaric” (para 1), meaning he was only half a barbarian.
In the very olden time there lived a semi-barbaric king, whose ideas, though somewhat polished and sharpened by the progressiveness of distant Latin neighbors, were still large, florid, and untrammeled, as became the half of him which was barbaric. (para 1)
The king is passionate and sometimes violent. His daughter is the princess.
This semi-barbaric king had a daughter as blooming as his most florid fancies, and with a soul as fervent and imperious as his own. (para 9)
Since the princess has such a passionate soul, she is just like her father. It is this passion and arrogance that leads her to guide her lover to either the lady or the tiger, depending on what you think she picked.
[With] all the intensity of the savage blood transmitted to her through long lines of wholly barbaric ancestors, she hated the woman who blushed and trembled behind that silent door. (para 14)
Chances are, the “hot-blooded, semi-barbaric princess, her soul at a white heat beneath the combined fires of despair and jealousy” chose to send her lover to the tiger. She is so like her father!
The princess and her father both possess similar natures.
In the exposition of "The Lady, or the Tiger?" the "semibarbaric king" is described in various ways, some of which suggest that he is powerful, self-indulgent, and inclined to having his dictates obeyed. As an illustration of these three character traits, the king has a public arena in which people are tried for their alleged crimes and judged by the "impartial and incorruptible chance." The king allows no other influences to affect the method of punishment but chance. Moreover, he does not allow for any objections to this method of deciding whether the subject is guilty or not.
The princess, his daughter, is much like her father. When she discovers that the man she loves has talked with the lady behind one of the doors, she is just as angry as her father has been upon learning of her lover. In addition, she, too, is described as "semibarbaric." And, she possesses a nature
...as blooming as his [her father's] most florid fancies and imperious as his own.
Because of this imperious nature like her father's, the princess has learned behind which door the tiger waits. Moreover, it is this barbarism in her, so much like her father's, which causes her to be present when the prisoner selects a door.