Of course no one questions that Amerindians were here long before the Norse. The controversy has been over the route they took. It was long assumed that the first colonists arrived when an ice-free corridor through the North American Ice Cap opened east of the Rockies about 13,000 years ago at the end of the last Ice Age. Mounting evidence of people present in the Americas before that has led to the proposal that the first colonists arrived via a coastal route, skirting the Ice Cap to the west. That’s what dtilque was talking about.
Has much DNA analysis been done of American indigenous people and compared their genome to those of North East Asia?
Yes. The genetic evidence indicates they came from Northeast Asia, but were isolated in Beringia/Western Alaska for several thousand years before moving south into North America proper.
The Anonymous Siberians sounds like a band name 
What makes Columbus’ discovery unique is that, after coming here, he went back and told everyone about it.
If you want to go that route, I’ll point out that Iceland is on the North American continental plate.
I think these are the best answers, because they highlight the meaning of the word “discover”. If you think that the OP is asking about the first outsider to reach this place, then you can debate about the Vikings and the Siberians et al. But (to me, at least) the concept of “discovery” includes not only finding it, but publicizing it.
There are stories that Irish monks, possibly led by St. Brendan the Navigator, settled Iceland before the Vikings did. In some versions, those Irish went all the way to Greenland or further.
Stories are not facts.
More to the point, those particular stories are embedded in a cultural context that contains many stories about finding mysterious land to the west. If you read medieval Irish literature, it is very hard to take Brendan’s journey, or anyone else’s, as literal, what with all the floating islands and giant mid-ocean pillars and guardian felines et cetera. Of course, most people don’t read Irish literature. Even so, how you could read Navigatio Sancti Brendani and not notice that it’s 99% religious allegory and 1% boat? The Irish should get some credit for discovering Iceland, but that’s about it.
Thag Simmons, but he didn’t live to tell.
But facts were once stories. They transitioned to facts because they were not dismissed out of hand. We have only “stories” that Cabot sailed the American coast. They are largely accepted because we presume that he had knowledge of Columbus.
What do you mean by “publicize”? The Vikings came back from Vinland and told everyone who would listen about a land far to the west where grapes and grain grew and where the winters were mild. The folks who were listening were mostly other Vikings, but I don’t think they kept the existence of the place secret.
Is there a concrete historical definition of “discovered” in this case? Because I think most historians would agree that the Americas were found and either visited or populated by a number of different people over the years without any singular “discovery”.
Columbus, as mentioned, is especially relevant because his personal discovery of the New World opened it to large scale European investigation and conquest.
Or as my Orgo teacher put it “Columbus wasn’t the first person to reach America, nor the first European to do so, but he was the one who published it. Research that’s not published might as well not exist.” Rodrigo de Triana (the outlook who yelled “land ho!”) was a minion, so in an actual peer-reviewed paper he might have been listed as a secondary author at best, maybe not even that and just an “our thanks to”.
Well, half of it.
Columbus promised a reward to the first person who saw land. He ended up claiming the prize himself, saying he saw a distant light at night when they were almost certainly too far away to see anything. Triana got nothing.
The academic analogy is still apt. 
Columbus wrote a letter describing his discoveries to his royal Spanish patrons. That letter was soon copied and spread to all parts of Europe within a few months in Latin and various translations. The discovery was highly publicized.
The Norse discoveries were mentioned in a few accounts that reached other countries in Europe. The main reason they did not become more widely known is that they were not regarded as being of any great significance. Vinland was just another land mass beyond Greenland that offered no great economic rewards or incentive to colonize.
If you make discovery relative to a subset of the population, it makes more sense - and applies to scientific discoveries. Who made the existence of the America’s known to Europe? Who made them known to Asia? Who made them known to Africa?
How did they know that, without mounting an expedition?