There's a lot of things which are tropes without any real serious timelines behind them. But yes you're right a Camnada colonized by Denmark would be radical different even if it followed the same colonization pattern. Denmark would run such a colony much more cheaply than the French ran New France or the Dutch ran New Netherlands, which would result in a radical different administration, it wouldn't be seen as a threat by the England or the British colonies like New France or even New Netherland were.
We can get a good idea from Denmark ran all their other wastelands (Norway, Iceland and Greenland) how Denmark would run Danish Canada. We would likely see a relative laize faire governance mixed with a hard enforcement of Lutheran orthodoxy at least until the 18th century, where we would see Moravians being allowed to go missionairing among the natives. At the same time while taxes would be low, the state would use trading monopolies to extract value from the colony. Military the colony would make use of a mix of a local recruited garrison and conscripted militia. Denmark would likely treat the natives different than they did the Greenlanders, but it would be far more likely to try to attempt to integrate them as subjects to the crown, it's not unlikely that Denmark would use a medieval style baptise and vassalise tactic mixed with a forced shift to European style way of life. The early settlers would likely be criminals deported to the colony, we would likely see them being settled as serfs rather than tenant farmers as in Denmark. Their lords would likely be expected to keep control over them and enforce royal degrees on them (like conscripting them). While the clergy would serve as civil servants keeping records, report back to the king and as representant of the king to the lords and freemen.