There didn't exist any "Norse people" in the first place, really. No more than there's a "Scandinavian" people today. It's a modern collective term for a number of peoples who had their own distinct identities and cultures.

Originally the word "Norse" just meant "Norwegian" in English. It was the British fascination with vikings and Norse mythology in the 19th century that turned the term into a general term for Viking Age Scandinavians. It was also that period that introduced 'viking' as a term into the English language. For most intents and purposes the whole image of vikings and Norsemen is thus a modern and foreign construct and perspective on Scandinavians. And many of the viking tropes and misconceptions that were spread at that time live on to the present day; horned helmets, berserkers eating mushrooms, runes being pagan and magical, etc.

The Viking Age itself has been defined in English terms; from the raid on Lindisfarne in 793 to the defeat at Stamford Bridge in 1066. So one might even question whether there ought to be a Viking Age from the Scandinavian perspective. The answer to that is yes, as there were still cultural shifts and changes that mark the beginning of the era: Raiding farther abroad, the rise of trading towns in Scandinavia and the Baltic, the development of Old Norse language from Proto Norse, etc, as well as the end of the era: The completion of the conversion to Christianity and consolidation of political power into larger kingdoms. But in those terms the Viking Age is more diffuse and stretches perhaps from 750-1100.

To put it simply: Swedes were Swedes, Danes were Danes and Norwegians were Norwegians, not unlike today. However in this period the kingdoms were not consolidated and were in fact broken up to different extents into petty kingdoms. Danes, Norwegians, Swedes-proper (Svíar), Geats (Gautar), Jamts (from Jämtland), Gutes (Gotlanders) were distinct ethnicities among the broader peoples, and these were further subdivided into peoples such as Jutes, Bornholmers, Trønder, East and West Geats, etc. My ancestors are Swedish in the present day sense but most hail from what'd be the petty kingdom of Finnheið in the Viking Age, which largely corresponded to the watershed of the river Lagan. They would've identified as Finnheiðar, not Svíar or Gautar. However this area was sparsely populated and many such peoples and areas are not spoken of much (or at all) in the sources (there are some rune stones from the area mentioning the term though)

The thing about the sources here is that apart from short and often formulaic rune stone inscriptions, there are no written sources at all from Viking Age Scandinavia. Stories were transmitted orally, writing with ink on parchment was one of the things that Christianity brought with it. The large bulk of the stories that we have are things that were written down 200 years later in Iceland, where the fornaldarsaga (tale of old times) became a popular medieval literary genre.

So the "Norse" culture that was constructed later based on these sources is based on an underlying assumption that everyone was like the Icelanders. Which we know is not really the case, and these assumptions have been increasingly challenged in recent decades. But it still permeates the scholarship (and will continue to do so to some extent). For instance, the Old Norse language is generally considered to split into what we term Old West Norse and Old East Norse during the Viking Age. (although many classifications exist) West Norse being the origin of Norwegian, Icelandic and Faroese while the Eastern branch became Swedish and Danish. There is also Old Gutnish which did not derive from Old Norse but is considered its own separate twig of the North Germanic branch of the Germanic language tree. But because of the massive amount of texts in 13th century Icelandic it is by and large the 'standard' form of Old Norse. Hence I write "svíar" even though it's "svear" in Swedish and Old East Norse.

"Language" and "dialect" are not clear nor very meaningful distinctions. Swedish and Norwegian are considered different languages today but are not really much more distinct than for instance High and Low German or Sicilian and Tuscan. There was a common Old Norse language (except for Gutnish) and it had local dialects but was by and large mutually intelligible during the Viking Age, and well after it, and to an extent to the present day.

There were also many common cultural influences and features and customs, and there was a certain concept of a northern ("norrœnn") joint identity. The language is referred to as 'northern tongue in Old Norse sources, and the Old Norse sources do variously show a distinction between fellow Scandinavians was made and other groups like Finns, Wends, "West men" (from the British isles), "South men" (Germany). If the older law of Västergötland for instance (circa 1200) is perhaps a hint at the attitudes here; killing a fellow Gautr is punished more severely than killing a Swede, which is still more severe than killing a Dane or Norwegian, which is nevertheless punished more severely than killing a foreigner of another nationality. Fellow Scandinavians had literally more value as human beings than other foreigners.

What we know of Norse mythology is as heavily based on Icelandic sources (mainly the Eddas) as anything. The idea that it was a single monolithic religion is certainly quite dead (to the extent it ever lived). Surveys (e.g. Brink of religious place names in Scandinavia show quite a bit of local variation in which gods were popular. There is an even more striking variation in local burial customs (Svanberg, Decolonizing the Viking Age), hinting at religious differences and perhaps even different conceptions of the afterlife. Svanberg goes farther than most there and would by and large completely discard the 'Norse' concept. The Gutes here had a quite distinct culture with for instance their prolific carving of picture stones. The Gutasaga, a short bit of local mythology written down in the 1200s has stories not attested elsewhere. The custom of carving runestones took on an entirely new character and popularity in 11th century Uppland in Sweden, where there are over 1,300 Viking Age stones in that region alone, as opposed to under 500 in the world outside Sweden. yet there are were also populated regions of presten-day Sweden that had no stones at all, or no stones of the 11th century type, like Dalarna, Värmland and Dalsland.

This is not to say there weren't many broad cultural similarities. Most of these gods were still known across Scandinavia, many myths were popular across Scandinavia. Some stories, such as Sigurðr Fáfnisbani even seem to have been common to the whole Germanic world. We also know there was plenty of contacts and intermarriage within the aristocracy and VIPs. (It continued long after the Viking Age was past) Unexpectedly, among this social strata you do have more common culture, with for instance royal burial mounds being a thing across Scandinavia. It is the perspective of this group that is expressed in the Sagas. It's about the doings of kings and their sons and so on, their alliances and marriages, it reflects their values and so on. So another way of looking at it, is that to some extent a single common Norse culture did exist, but within a small segment of the population.

Still, a term like "Norse" is useful if not outright necessary in contexts outside of Scandinavia. Continental sources tend to speak of "Northmen" and other unspecifci terms, English sources often use the term "Danes" for Scandinavians regardless of whether they were actually Danish or not. And raiding parties (or líð) and groups of settlers in places like the Danelaw and Normandy are likely to have had members of various nationalities.

But on the whole (and contrary to common belief) the cultures here were not unified in a single "Norse" culture but rather fragmented into more local cultures, customs, laws and cults. (and having your own provincial laws, as they did, seems to have been an important part of the local identity by the 11- and 1200s) The various differences do not necessarily follow neat boundaries either; neighboring regions may have one thing in common but not another and so on.

Ultimately, there's not so much we can do about the fact that our sources are Icelandic by a huge margin, other than be aware of that fact and take it into account.