I haven't seen this particular fallacy described elsewhere. It happens often enough that I was thinking it ought to have a name, and I was thinking of calling it the fallacy fallacy, but that refers to a different fallacy, and it seems to me it's better to leave it unnamed, in light of its nature.
So what is it? It's the belief that calling out the name of a fallacy is an argument. "That's a straw man" or "That's a No True Scotsman" are things you'll hear all the time in debates, but it's almost never a good idea to use the name of a fallacy in an argument, for several reasons.
First, it's very often misused. Knowing the names of fallacies is no guarantee that you actually know what makes them invalid. Many people seem to think that ad hominem just refers to name-calling, for example, or will call out "No True Scotsman" if you try to define a term in a way they don't like.
Second, it's lazy. Even if you do happen to properly understand what the fallacy is and why it's a fallacy, simply naming it is seldom an efficient shortcut to making clear your opponent's error, in part because (remember the first reason) there's a good chance your opponent either won't, in which case you're wasting words.
Third, it very often leads to completely unnecessary side-arguments about the definition of the fallacy itself. "What? That wasn't an ad hominem! Ad hominem is when I say you're stupid, therefore your argument is wrong." "No, ad hominem means a personal attack!"
Fourth, it's likely to be seen (often correctly) as showing off, an attempt to telegraph that you know something about the technical aspects of argumentation and therefore are not to be messed with, you master of logic you.
Finally, it's almost always completely unnecessary. Remember that a fallacy is a flawed argument, an error in reasoning that makes it vulnerable. You don't need to name the flaw in order to attack it. For example, recognizing that an ad hominem is a form of non sequitur where the premise ("You're stupid") does not lead to the conclusion ("therefore, you're wrong"), you can note that you do not need to refute the premise. "I may or may not be stupid, but stupid people can be right and smart people can be wrong. Show that my argument is wrong."
That is, by the way, why it's definitely useful to have names for the various types of fallacies, so we can discuss and analyze them and learn how to recognize and counter them, and to avoid committing them ourselves. Shop talk about rhetoric benefits greatly from having terms of art like these. But naming them in the middle of an actual argument is rarely a good move.