That's one of the signs that this truly is a misogynist phenomenon
It's a catch-all terms whereby -
If a woman protests at being called Karen, she is then labelled 'a Karen' for doing so.
If a woman who is actually called a Karen feels sad about her name being used as an insult, and voices frustration over discrimination she has faced because of her first name, then she is also typically a Karen and should check her privilege. She needs to learn from this lesson and live with the discomfort of her first name to be in some kind of solidarity with people of colour who have faced racism (I've read an article that promoted these ideas a while ago).
If a woman called Karen actually changes her first name because of all of this, then she is also a Karen as she has the privilege of doing so.
If a woman called Karen doesn't care what people think, then she is also typically a Karen has Karens apparently insensitive and don't care about the opinions of others. By being nonchalant about the whole thing, she is displaying her privilege.
And btw @Lndnmummy @AnObserverInThisDarkWorld I've been called Karen for calling out sexually questionable comments towards a young woman in a Facebook group. It's is applied like this.
Rest assured that this term will also come for you, and if you don't see that, then you are quite frankly dim.