Insofar as the concept of nationality existed in the early modern period, Norwegians were considered a different people to Danes. Even after Norway was turned into a province in 1537, the country and its people were considered inherently distinct. This started as a political/geographical signifier - people born and residing in the region known as Norway - but eventually became a cultural signifier inspiring agitation against Denmark and for Norwegian advancement.

Before the inception of the Kalmar Union, Denmark and Norway had already existed for centuries as separate states, and the two countries' elites had built national mythologies as part of royal consolidation efforts. After the union, the Norwegian and Danish aristocracy stayed apart. The king in Copenhagen was considered to be the king of two countries separately, although one was clearly superior. The elites were not intermarried, and the Norwegian nobles were barred from the privileges enjoyed by their Danish counterparts. This distinction led the elites in the two countries to sponsor art and histories drawing on separate traditions, creating separate national mythologies.

As for the common people, whether they considered themselves "Norwegians" is more difficult to gauge. After 1814 a popular historical tradition held that the Norwegian national identity was kept alive for 400 years by our fiercely independent peasantry. That's a little hard to evaluate, though. What is known is that Norwegian peasants were hostile to, and often sabotaged, Danish administrators. Feudalism was never meaningfully established in Norway, and farmers were never made serfs like in Denmark. There were riots and resistance to the Danish imposition of Protestantism from 1537, though this died down as we were allowed to gradually phase out folk and Catholic traditions. When students were sent abroad, the prefixes Danus and Norvegus are probably the most direct proof that there was a "national" distinction between the two.

The last clue is the professed attitudes of the Danish kings. In royal decrees and dispatches, the kings always recognize the fact that Norwegians had distinct characteristics that marked them as different. They were more hardy, better sailors, generally a people they wanted to keep in the union but who they did not in any way consider "Danish." Always the Norwegians were referred to as a clearly distinct people.

After 1537 and the abolition of the Norwegian council of state, the "Kingdom of Norway" ceased to exist, but due to the separate climates and histories, Norway's geographic expanse and the failure to meaningfully integrate Norwegian elites, no common identity was ever forged.

This is mostly cribbed from this article by Øystein Rian, and Norsk historie 1536-1814 by Ståle Dyrvik. The former is a good read if you know Norwegian.