Basic Trope: Princesses make any given series that much better just by the virtue of being princesses.
Straight: Princess Mary is one of The Hero Bob's Love Interests, as well as the nominal ruler of their homeland.
Exaggerated: Princess Mary Sue is absolutely revered by all who lay eyes upon her and lauded as the most pure, beautiful and amazing member of the royal family they've ever had the pleasure of serving.
Downplayed: Mary mentions that she would be a princess, if her country still existed. It gives her a bit of exotic background, but has no bearing on the plotline.
Justified: The show is an adaption of a fairy tale or a series about a royal family, so it would be odd if no characters were princesses.
Zig Zagged: Princess Mary is introduced as the heir to the throne of an ancient kingdom, but we eventually find out she was just being delusional and that such kingdom hasn't ever existed. But then we find out she was really a princess of another kingdom... Except, she technically isn't because an usurper queen is on the throne right now. So the heroes have to defeat her to reclaim Mary's right to the throne.
The show is set into an alternative medieval period where royalty was discarded in favor of democracy.
Alternatively: There is royalty. There just doesn't happen to be a princess at the moment.
Enforced: "We needed to establish a link with the royal family early on to justify the king's favor that occurs later on; so we decided that Mary joining the adventurer's group would be a good way to do it"
Lampshaded: "To make it clear princess, don't expect to receive special treat from us just because of your status; we are adventures, not servants" "My poor feet are aching... Would you be so kind to carry me on your back for a few miles?" "Yes, your highness!"
Invoked: Convinced by her friends, Alice masquerades as Princess Mary with the hope of gaining the favor of a friendly king.
Exploited: Con woman Mary dresses up as a princess as the local people love princesses, and will fall for her plan.