There is however a big difference between the Danish decision now and the Swedish decision to end male primogeniture.

The first one is not a constitutional change in anyway. As a matter of fact, it is stated that the four children keep their place in the line of succession.
In the Swedish case you mentioned, it was a change of the Constitution so yes, obviously politicians had to be involved.

The way titles are afforded to members of King's (or Queen's) family differ from country to country, but in this instance it seems it truly is a royal prerogative of HM The Queen to decide who is and who isn't and HRH and Prince(ss) to Denmark (as I understand only the children of the CP are Prince(ss) OF Denmark).