There are a lot of questions about which fallacy something is on this site.
Many times, the argument in the question is not actually a fallacy, but something that the questioner doesn't like/agree with or some other behavior that's not a fallacy.
Is it a fallacy to call every problem in a discussion a fallacy? If yes, which one?
Calling something a fallacy when it is not is of course a mistake.
And, you're right: sometimes people will try to reject the conclusion of an argument by pointing out a fallacy with the argument, and that is called the Fallacy fallacy: just because I make a bad argument for some claim $ does not mean that claim X is suddenly false.
Indeed, sometimes people (especially after taking a first course on fallacies!) will get too caught up in trying to point out fallacies. To them, pointing out fallacies has become a weapon to try and win arguments, rather than that they try to improve arguments in our search for the truth. For example, argument that they attack may actually contain important observations, points, or kernels of truth, but by focusing on the weak spots of the argument, they think they can reject the whole argument, including its conclusion, and so they'll throw out the baby with the bathwater.
I think you may enjoy this comic "The Adventures of Fallacy Man"
SHORT ANSWER
To call everything a fallacy isn't a bad argument, it is bad argumentation, technically, so no.
See this related post here fallacies regarding invoking fallacies.
LONG ANSWER
It is true that many people engage in what I've heard called fallacy mongering, which is the tendency to reduce all of your opponents' disagreeable points to fallacies, often with little regard to the actual definition of fallacy or the fallacy in question. Often times, fallacy mongers will accuse one of a fallacy in anticipation of a point before it is even made, often hurling the name of the fallacy exclusively:
You> Do you understand that scientists accept evolution as fact?
Them> Appeal to Authority!!!
You> Are scientists' expert opinions not weighted more heavily in a court of law or by society?
Them> Appeal to Popularity!!!
You> Okay, but doesn't the publication known as the US Federal Rules of Evidence show that scientific authority goes beyond mere percipience in specific matters?
Them> False Attribution!!!
In this slightly hyperbolic example, just by invoking a proposition in which experts play a role, the opponent in the argument believes they are refuting your conclusion that hasn't even been made. Of course, this is a sign that a partner in argumentation is often unaware that argument has widely accepted rules, such as recognizing burden of proof, staying confined to the question, acknowledging enthymemes, deciding upon acceptable logics, etc.
While you cannot stop others from fallacy mongering, you can learn to recognize it, and avoid it yourself by first and foremost recognizing:
Francis Dauer in his Critical Thinking asserts a fallacy is:
an erroneous but frequently persuasive way of being led from a reason or circumstance to a conclusion.
But I find it better to recognize necessary and sufficient criteria and rely on three by T. Edward Damer in his Attacking Faulty Reasoning. An informal fallacy is an argument that is built in part or whole on:
- Unacceptable propositions. Can the truth of the proposition even be determined, for instance? You can't prove that many angels dance on the head of a pin. Do colorless green ideas really sleep furiously?
- Irrelevance. Is your claim or your conclusion pertinent? Is something non-sequitur? And your point is?
- Ungrounded: Even if a premise is acceptable and relevant, does it actually imply the conclusion in some manner? Can you get there from here?
Your question could use examples.
Either an ad hominem [what relevance is my gauche shirt to this discussion], or, depending on the seriousness of the discussion, the fallacy of irrelevant conclusion [what relevance is it that I think it's a nice shirt] may apply, to one or other of the parties.
I've been in lots of discussions where someone is not interested in their soundness of their reasoning, just want to win the argument. It is not pretty, and I'd guess is what is going on -- in the discussion/s the question refers to.
Being so is very important in discussing philosophy: more so, even, than being able to concede defeat. Both those fallacies fall under abuses of 'truth' and missing the point, anyway. Perhaps that's where analytic philosophy is successful: that is avoids those vices -- in italics -- via an ability to clearly state arguments.
Unless you mean the over-use of 'fallacy' to apply to statements that don't incorporate any reasoning at all. In which case, forgive them: 'fallacy' often just means "false" in the popular imagination.