Well, you said it yourself, the NHL is not the NBA lol.
Context for why they fought in this game:
The red white and blue team is the New York Rangers. The red white and black team is the New Jersey Devils. They are indeed long time divisional rivals. The hatred is real. Also for context, the last time these two teams played, a Rangers player injured a Devils player with a nasty hit; he was challenged to a fight then but refused. So the tension had already built up.
Context for why they were allowed to fight:
Fighting has always been allowed in hockey. Big line brawls like this used to be more common even into the 2000s. They have become rare starting in the 2010s when the NHL meta shifted from "have dedicated face punchers fill out the bottom of the roster because they are scary to play against" to the new meta of "have skilled players even at the bottom of the roster because winning games is more important."
If you're wondering why the officials don't break them up immediately, it's a matter of practicality. There are only 4 on ice officials and there 10 angry dudes punching each other. Best to let them tire themselves out before intervening. Usually they will reach a point where they stop on their own. Also, the referees never break up scrums or fights even if there are only 2-3 guys fighting - this is so they can accurately pay attention to all of the infractions and make the proper penalty calls after it is all settled.