This was frequently cited by media or other commentary as “childish” behavior. Of course many children, not yet fully developed have no better response to being wounded than responding with “well, so are you!”. As I understand from the books I have read on the topic, narcissists fail to develop a broader sense of self, leaving them stranded in this kind of childish mindset. It’s got to be a kind of torture.
Or, as Trump would say: Sad!