They were not the original inhabitants (although some Scandinavian historians believed that in the 18th and 19th centuries). We now know the peninsula, even the northernmost parts have been continuously inhabited since the ice receded after the end of the last Ice Age (some 9000 years), and the people living at that time could not really be claimed to be ancestors of one group or another; there have been multiple waves of cultural changes and immigration since. Both the Finno-Ugric languages they speak and the Indo-European (Germanic) languages spoken by the majority populations today arrived thousands of years later. (and one can't really reduce a people and their origin to language either)

But the designation as 'indigenous people' isn't really about who was there first. It's about the fact that the Sámi were the people living in these areas (Sápmi/Lapland) when the newly-consolidated Scandinavian christian kingdoms claimed the areas at the start of the Scandinavian Middle Ages (1100s and 1200s). That made them an ethnic minority within the kingdom but a majority within Sápmi, which they remained until the 19th century when mining and other industries attracted migration there. (and they're still the majority if you exclude urban centers)

It was not really a conquest; there was no Sámi state that was invaded. Some missionaries were sent by by-and-large the Sámi were left to their own devices during the Middle Ages. But - in a key difference to colonization - they were considered citizens. Which also carried the obligation to pay taxes. (and an obligation to be Christian, but that wasn't enforced until the 17th century) But they also were allowed certain rights the majority population did not have; Sámi in Sweden (the including Finland) were permitted to hunt elks (moose) and other game, that was a privilege of the nobility in the rest of the country.

By the 17th century, Sweden is a staunchly Lutheran country. There's no freedom of religion; citizenship is not distinct from membership in the state church, attending church is mandatory. Catholics and Jews were banned from the country (with some exceptions made for foreign dignitaries) Converting to Catholicism or even another Protestant sect like Calvinism, was considered treasonous and punishable by death, and over a dozen people were executed for it in that century.

And so in that century, churches begin to be established in northern Sweden and Finland, the Sámi who were not already Christian were forced to convert and also to be registered in the population census. There's no denying it was a forced conversion though, but mainly by threat; people on trial for having engaged in Sámi religious practices who confessed and repented at trial were not punished, so only a single Sámi man is known to have been executed in that century for engaging in pagan practices. (with a second suspected case) Considering the hundreds of completely innocent Swedes and Finns out of the majority population who were executed for practicing witchcraft in that century. Obviously, conversion was destructive to their popular religious traditions, although some shamanistic practices survived well into the 19th century. On the other hand, it did benefit their language; as preaching in the vernacular and having people understand the religion in their own language were core tenets of Lutheranism, the Bible and Luther's Catechisms into Sámi languages, of which very little had been recorded in written form previously.

At the same time, as they were now included in the census, the Sámi were given parliamentary representation. (which was in fact not that popular - as members of parliament had to pay for their own travel to Stockholm and no translators were provided for them)

Things got worse in the last couple centuries. The 19th century saw the rise of Nationalism across Europe, and 'scientific racism' and racial theories. These racial theories typically equated 'race' with language, as if language were the main indicator of heritage. (These ideas are shockingly prevalent to this day considering there's little truth to them. DNA analysis shows that the Sámi are not genetically closer to Hungarians than, say, Slovaks, even if the former speak a related language and the latter don't)

To the racial theorists of the time, the Sámi (and to lesser extent Finns) were an inferior, Asiatic people. They - like Native Americans - were treated badly for instance by the Ethnologists of the era, who saw no need to ask permission to desecrate burial sites in order to skulls to come up with phrenological ideas about the "Sámi skull shape".

To the nationalists, Sámi culture and identity was a threat to national unity and the idea of one people, one nation. There are large differences between the countries here, in the extent Sámi identity and culture was supressed/tolerated.

Norway had been under Danish rule for centuries and was forced into a union with Sweden in the early 1800s. Although they had far-reaching autonomy within it, nationalism in Norway was spurred by desire for full independence, and there was an active program to 'make Norway more Norwegian' - not just regarding the Sámi but for instance in the creation of a new standard for written language (Nynorsk) based on the dialects of Norwegian that had the least Danish influence. Norway adopted strong assimilation policies; Sámi languages could not be used as the main language of instruction in any schools; on the contrary, special schools were created in Norway to assimilate Sámi children into Norwegian society. Further, a law was passed in 1902 limiting the sale of land to Norwegian citizens who could speak, read and write Norwegian and used it on a daily basis. This was only repealed in the 1960s, when attitudes began to soften.

Sweden was more tolerant; no such laws restricting rights by language existed. Schools with instruction in Sámi languages were created, the policy being "Laps should be Laps." - but the idea of "Laps" here was rather narrow and stereotyped; only children where both parents were reindeer-herding Sámi were considered Sámi and entitled to school instruction in Sámi. Attitudes to expressions of Sámi identity and culture (e.g. wearing of their traditional clothes) was also less hostile in Sweden compared to Norway.

In Finland, attitudes were even more tolerant. Not least since (as said) the same scientific racist theories that deemed Sámi as inferior and Asiatic had also claimed the same about Finns. Which is not to say that kind of thinking was completely rejected in Finland, but within that framework Sámi were thus a cousin if not brotherly people, and afforded relatively more respect.

After WWII, attitudes changed in all the countries towards greater tolerance and inclusiveness.

The short of it is that the Sámi were never treated anywhere nearly as badly as the Native Americans. Most significantly they were - from the start - considered citizens. But their culture, language and identity was not respected, particularly in the era from the mid 19th to the mid 20th century. They still have many issues where they can find common cause with Native Americans when it comes to things like preserving their identity and culture and traditional ways of life.

Well, first: The Sami are indigenous to Sápmi. They were living in those areas first, by all accounts.

Second, it's mainly a political designation. The Sami were largely left alone until the 19th century, at which time the governments in question started making active efforts to assimilate and integrate them. Their culture was suppressed (e.g. they were forced to attend schools where they were banned from using their own language), and they were subject to discrimination and prejudice. In those respects, their situation was highly similar to that faced by indigenous populations subject to colonization.

It's not like they were merely an ethnic minority who had their own culture and perhaps suffered discrimination, but were nevertheless an integrated part of the society. In the late 18th century 'civilized' society was still largely finding out who these people were. The famous biologist Linneaus made an expedition up to Lapland, describing Sami customs and posing for pictures in their dress; they were more subjects of anthropology than subjects of the crown at that point, and viewed more as primitive indigenous peoples of foreign continents were than as fellow Europeans. Jean-Jacques Rousseau (who corresponded with Linneaus) wrote in Emile that "Neither the negroes nor the Laps are as wise as Europeans."

So their recognition as 'indigenous people' here simply shouldn't be taken to mean that they're 'more indigenous' than other people within the same present-day national boundaries. It's a recognition of their right to their own language and culture, and an official acknowledgement that the governments in question don't have a right to (for example) wander in and tell them they can't herd reindeer where they've been doing it since time immemorial.

(While I feel I've argued enough about this recently, 'Norse' is not a demonym. You're really talking about Norwegians, Swedes, Finns and Russians here)