I've seen some headlines about countries separating from the monarchy now.
Not familiar with the details.
Is it just an opportunity for change, or something about the King?
I think it's a bit of both. Queen Elizabeth was much respected for her work ethic, for her years of service to both the UK and the greater Commonwealth, and for the fact that she was VERY diligent about remaining politically neutral (as the Monarch MUST be seen to be).
Charles on the other hand lost a lot of his popularity (rightly or wrongly) as a result of his marriage breakdown with Diana and it's aftermath. There was also the issue of his political activism. It's fine for him to openly talk about his views on modern architecture and saving endangered species, another thing entirely to involve himself in political debates. I think William understands this better than his father did at a similar age. That said Charles has said all of the right things since becoming King so we shall wait and see.
Then there are the effects of historical revisionism which has been on full display since the Queen's death. It has been very obvious that most people in the media do not understand the role of the Crown in a British style Constitutional Monarchy. Unfortunately most people seem to get their information from twitter, celebrities or activists parading as journalists. The result is a growing desire for the Crown to be ditched and replaced by a President without understanding the possible consequences that this could have to the political system.
In Australia we had a referendum in 1999 on whether or not we should become a Republic. Most people in opinion polls leading up to the referendum supported the idea but there was disagreement as to how the President would be elected, what powers he would have and how (or even if) he could be dismissed. In the end the referendum resulted in Australians saying no to the republican option presented to them. The current government has reignited the debate and effectively promised another referendum in 2024.
At the moment (excluding the UK) there are fourteen countries who all share a nominal head of state, that currently being King Charles III. These countries (plus the UK) are called Commonwealth Realms. I would think that over the coming 5-10 years a number of these Realms will decide to become Republics.