I really like your question and I share your frustration at the way people overuse and misuse references to fallacies. For some very unfortunate reason there are many people around who think that logic is the study of fallacies and who think the way to assess an argument is to hunt for fallacies in it. We get endless questions on this site of the form, "what is the name of this fallacy..." and the answer I want to scream at the questioner each time is: it doesn't matter - forget about names of fallacies and just learn to think critically.
I suspect the reason this happens is that people think of logic only in the context of an argument with other people, and they treat an argument as a kind of game in which if you can name a relevant fallacy you win. The most important reason to study logic is to improve your own reasoning and avoid errors. To this end there is some value in learning about common fallacies, so you don't commit them yourself, but it is important not to get hung up on it.
Claims of informal fallacies are in effect a kind of reasoning by analogy. They amount to a claim that an argument is defective because it resembles in relevant respects some paradigm case of a defective argument. Reasoning by analogy can be helpful and persuasive, but there is always the hazard of stretching the analogy too far. Real-life arguments are often subtle and nuanced and depend on considerations for which no obvious relevant paradigm is available.
As a matter of practical advice, I suggest that if someone objects to an argument on the basis of an informal fallacy, challenge them to say what is wrong with this particular argument without appeal to the general category of the fallacy. What is it about this argument that makes them think it resembles the paradigm? Answering that question will give you an opportunity to respond.
When you ask, "...is there a name for... a tendency to over-identify informal fallacies..." you seem to be falling into the very error that you criticise in others. Although if we understand the request in a strictly ironic sense, then I propose we define a new fallacy as follows: The Lazyman Fallacy := the fallacy of supposing you can defeat an argument just by appealing to a named fallacy.