In the Disney movie Moana Moana has an argument with Maui about her princess status. It got me thinking. People often point out about Pocahontas that "native American princess" is a completely culturally nonsensical concept. They was no such idea in her culture. Is it the same with Polynesians. I recall reading somewhere that the daughters of Maori chiefs had a special position, titles and status but can't recall the details. And I don't know if the rest of the Polynesians were similar and Moana isn't Maori. So if I am a Polynesian girl and the daughter of a chief what exactly is my position in the tribe. How much could it be considered that of a Princess? More generally how hereditary was authority and status in Polynesian societies.
I can only speak for the Samoan system (fa'amatai), but you're getting into very complicated waters. The only hereditary "Western monarchical" systems prior to European contact were Tonga and Hawai'i, so I'm not sure there's a direct analogue to a "Princess" or even a "king." The Samoan word for a paramount matai over a group of islands is "tupu", but that's not really the same thing.
Each family (aiga) in Moana's village would have a matai (chief) of the family, selected by the extended family. The title (effectively a name like Tu'i Manu'a or Tupua Tamasese) would stay with whoever was made matai and aren't necessarily hereditary. So the families that make up Moana's village would each have a matai who would make up the "fono" or village council, presided over by the ali'i or paramount chief of the village (in this case Moana's father), determined by intricate and complicated rankings of the various matai titles. Both men and women have equal claim to the title, but customarily men are the vast majority of matai. Complicating this is the role of the "tulafale" or speaking/talking chiefs, who speak on the behalf of the chief of the family. There's also the nu'u tama'ita'i, which is the highest-ranking female of the village, though not necessarily the daughter or wife of the matai.
Moana is the daughter of an ali'i, so in rank she would be called taupou and have several privileges and responsibilities. She would have a retinue of other umarried girls and, so long as she was a virgin, she would dance the tauluga at village ceremonial events. But she would have no outstanding right to being the ali'i upon her father's death or retirement. EDIT: That being said, stopping the end of the world would certainly qualify her for her own title separate from the ali'i title of her father.
There was a brief attempt in the colonial days to install a "King of Samoa" that would serve as a head of state for colonial powers to interact with, but that failed somewhat miserably and sent the islands into a series of brief civil wars.
Source: Living and working in Samoa.
I'd also check: The Making of Modern Samoa and A Short History of Western Samoa, both by Malama Meleisa