Nation states are a modern idea, true. That doesn't mean nations - in the sense of large groups with a sense of shared identity - didn't exist. An Englishman in the 14the century knew he was an Englishman. A Fleming of the same period would have found it normal to be part of France but would never have considered himself a Frenchman. What they didn't have is the modern concept of nationalism: the idea that nation and state somehow have to coincide. But they knew who they were.
Of course people have always identified with some group. But nation states made this group much, much larger. Before, the group might've been just a village, or Gilford the Genericknight's holdings. From what I understand, the closer people were to being able to affect nation-level politics, the more they identified with their nation. Which wasn't the case with common farmers, most of whom were serfs(effectively slaves bound to a plot of land) in many European nations during the Middle Ages.
In general, Skyrim and other fantasy RPGs give a very strange picture of Medieval life. Common people weren't generally free to move around and pick their own professions. They could own land, but had to spend a significant part of their time working on their lord's land, or otherwise work for them. In addition, they had to pay some of what they produced as taxes for the lord in exchange for their "protection". Much of medieval society revolved around supporting the Feudal lords. They were the state and people were obligated to serve them. Lucky serfs could buy their freedom, but it wasn't that common. In 1086, only 12% of William the Conqueror's subjects were freeholders, i.e. non-serfs and non-nobility who owned their own land.
That being said, I don't know enough of Scandanivian history to know when "Swedish", "Danish", etc. emerged as national identities.
The birth of Sweden as a distinct nation could be considered to be the point where Gustav Vasa broke it away from Norway-Denmark in around 1520 or so. The common Swede was never a serf, unlike most of the population was in many countries in medieval times, so their belief in being able to affect their own destiny would've made the the transition towards a nation state faster.
Not sure about Norway and Denmark, but I gather that generally the history of the three nations has been a constant mix of people from each conquering the other for a while, until the tide turns, and they've been sort of a single, quarrelous nation through most of it. Hell, most of the area that's now Danmark, Finland, Iceland, Norway and Sweden was even ruled by a single king at one point. See Kalmar Union.