My feeling is that the Polynesians would be divided among many many tribes like the American Indians in North America.
We can assume that the initial colonization of your Hawaii would be from only a few groups in the first centuries, likely from Samoa, Fiji, Tonga, Tuvalu and possibly the Marshalls and Kiribati. There may have been as many as a dozen significant colonizations, or perhaps as few as two or three principal ones.
From there, we can assume a high degree of linguistic and cultural coherence. This will diverge rapidly. But by and large, most of the cultures will be closely related, and languages will be close. The only wild card might be micronesian or melanesian influences.
There would likely be quite a bit of dispersal and cultural and linguistic divergence, but it would be nowhere near as heterodox as the tribes and languages of Africa, the Americas or even Australia. The time scales of these places are all in the tens of millenia. Hawaii's pre-contact history is maybe two millenia, give or take.
So my question is in what important ways are Polynesians different from North American Indians, and I don't mean by physical appearances or their root language. North American Indians were a bunch of nomadic tribes often at war with each other, they practised limited agriculture and their meat was usually obtained through hunting, I believe the Polynesians weren't much different than that, they were an Island seagoing culture that obtained much of their food through fishing.
Hmmmm. No offense, but you might want to do a bit of research on Polynesian and Micronesian cultures and history. A bit more nuance on American Indians might help as well.
Polynesians were indeed skilled fishers, and both inshore and deep water fishing were important economic activities and key parts of their diets. But they also raised pigs and chicken, and they also cultivated land. In many polynesian islands, there were vast agricultural works and pretty much all arable land was cultivated and irrigated. The staple of the diet was not fish, but agriculture.
The only Polynesian model that resembles your view of North American Indians were probably the Maori of New Zealand's South Island. What happened there was that the Polynesian agricultural package didn't work at all. The Polynesian package was tropical. It barely worked in the North Island, requiring a new domesticated crop and considerable adaptation. But the South Island was simply too temperate, tropical crops didn't grow.
The South Island Maori therefore were forced to switch to a hunter/gatherer lifestyle, with a very limited (if at all) agricultural component. They then proceeded to hunt out the local game, the Moa, experienced a population crash, and re-established themselves at a relatively low level.
Now I don't believe Polynesians raised livestock before the Europeans made contact with them, they only brought themselves with they came to Hawaii.
Well, as noted, they did have several domesticated animals that they used for meat, eggs, feather and leather. However, they had no draft animals obviously.
I think there might be some animals they might possibly domesticate. Do you think a flightless bird the size of a horse is within the range of possibility? How about the idea of bird riders?
The modern giant Birds - the Ostrich, Rhea, Cassowary, Emu, Moa and Aeropynis, found (or recently extinct) in Africa, Australia, South America, New Zealand and Madagascar are all Rattites. Most, if not all, of the giant flightless extinct birds seem to have been Rattites. I assumed for a while that Rattites were a catch all category for a group of unrelated species. I've since learned however that current thinking is that the Rattites are all products of a common ancestor which evolved during the age of dinosaurs and managed to survive the extinction event, producing descendants in the Southern continents.
Will you get Rattites in Hawaii? it depends on where your Hawaii comes from. If it develops in isolation, without ever having been a part of the supercontinents, then no Rattites. If its a breakaway part of another continent or supercontinent, then its possible.
In OTL timeline, we did see the evolutlion of giant flightless ducks in Hawaii. But 'giant' is a relative term, they weren't much bigger than big turkeys. Conceivably, without competition, empty niches and a longer period to evolve, we might have seen larger and more diverse specimens.
As for riding... riding actually shows up pretty late in stock animal domestication. The earliest uses of domesticated animals for labour seems to be in carrying burdens or pulling loads - thus we have dogs pulling inuit sleds or Indian travois. We have Llamas and Oxen and Reindeer carrying packs. Oxen and Water Buffalo pulling plows.
Actual riding shows up with horses, camels and somewhat with moose and elephants. Even then, it appears that horses were used to pull carts and chariots as much as ride, and pulling loads or carrying burdens may well have been the first uses.
It's possible you'd get a big riding bird, if you care to design the species. Go for it.
Birds are after all the closest living relatives of the dinosaurs, it seems reasonable to suppose that an isolated continent might evolve very large varieties of flightless birds. How about a "Tyranno-raptor" for example, basically a huge carnivorous bird that approaches the size of a T-Rex, though something that size would need some other type of prey than humans.
You might want to read up on the carnivorous terror birds of of North and South America, notably the Phororuscids. Also, check out the Australian Demon Duck of Doom. It may give you some ideas.
Now if the Polynesians were on Hawaii for 1300 years, would they have had time to kill off all the continent's population of tyranno-raptors? would they even have conquered the entire continent is such a short span of time?
I'm thinking more likely double that time span. But having said that, it's very much up in the air. You'll likely have extremely variable population densities. And likely a lot of the original habitats are going to be intact. I'd say its likely that a lot of the original flora and fauna survives into the modern era.
But then, I'm being a bit heretical. The consensus is that invasions of this sort invariably result in Megafauna extinctions - as seen in Australia, the Americas, Madagascar and New Zealand. But that's another argument.
China and India are huge populous countries that existed long before Europeans ever got their, I'd say in the sense your talking about, the first European to "colonize" India was probably Alexander the Great, but he didn't colonize India in the American sense of the word. I don't really get a sense of Polynesian Hawaii being a really ancient culture like China or India, you realize of course that both Buddism and Hinduism are much older than Christianity, and China and India were never really British in the way that the North American colonies were.
Well, the North American British colonies were British through settlement of Europeans and displacement or extermination of the Aboriginal peoples.
But even in the Americas, that was hardly cut and dried. A large portion of Mexico's population for instance is Indian or mixed Meztiso. South Mexico and Guatemala still have huge Mayan populations. Natives form a large part of the populatlions of Venezuala, Columbia, Peru and Bolivia. Even in New Zealand, the Maori have proven numerous enough and resistant enough that they're an ongoing political and social factor.
Your question for Hawaii is whether European colonists can successfully displace an extremely well established native population. By and large, in Africa, in the Middle East and in Asia, that has not been a good bet, and even in Latin America the results were so so. For the most part, the record with existing Polynesian cultures in our time line is that they survived European conquest and avoided displacement by settlers, except in outliers like Easter Island, New Zealand and OTL Hawaii.
Will displacement and settlement occur in your Hawaii? It's a toss up. If the population is well established and comparatively sophisticated, its hard. If the population is extremely vulnerable to disease, it might be easier.
I think its likely that Hawaii would be colonized in the same era that Australia was, these are the last few continents to be discovered by Europeans, they would naturally want to colonize those land masses that are closest to them first unless..
Discovery and even trade is not synonymous with colonization and conquest. Pacific Islands were known for centuries before the European flags started getting planted on Islands. That's why Germany had Pacific Island possessions - it didn't even exist while these places were being discovered.
While the history of the Americas is one of outright conquest, the history in Asia and Africa was one of trading networks and trading rights, spheres of influence and commercial ventures. The traditional European Colonial/Conquest in Africa and Asia was really a 19th century thing.
So basically, you've got two models to work with - the Americas model and the Asia/Africa model. Your Hawaii doesn't quite fit either one, but I'd suggest its closer to Asia/Africa.