Does invoking sophism in an argument be considered a logic fallacy ?

Can’t tell you how many arguments I’ve had that is over something objective. Does this medication work? Is god real? Any new age claim or paranormal claim. Once I back someone into a corner I get told the legendary “Everything is subjective”. Or the old Nietzsche quote of “Their are no facts”. I just replay with can you prove that?  What’s even the point of having a disagreement with someone if whenever you been pushed into a corner you just say in some sense that only you know your mind is sure to exist???? 

Most of the time, this is a failure to be an honest interlocutor. It is basically refusing to have a reasonable discussion. Other times, it is a person's cognitive dissonance kicking in, which is often a good thing. You may not think you are getting anywhere, but at some level, progress is being made. The key is to catch the person on special pleading . For example,

Person 1: Does medication X work? The vast majority of reputable studies seem to claim it does.

Person 2: No, it is a sham by the medical community.

Person 1: Where is the evidence for that?

Person 2: The evidence is being suppressed by Big Pharma and the Deep State. That's why there is none.

Person 1: How come they aren't suppressing the thousands of other non-pharmaceutical treatments studies in scientific journals?

Person 2: Because... Illuminati... Obama... (frustration)

People are stubborn, but for many, reason ultimately prevails. Don't give up.

This seems to be a Thought Terminating Cliche; an attempt to divert.

Subjectivist fallacy is another option, in addition to what Dr Bo already (accurately) said.

I like to call it "retreat into subjectivity" - a person is more than happy to argue a common reality up to a point, but past that point, they "retreat" by creating their own reality - a personal 'realm of truth' that's totally arbitrary, defined on their terms, and established such that they can never be wrong. They then assert a claim and deny all evidence to the contrary by constantly appealing to subjectivity, effectively substituting the common reality for their own.

It's akin to a chess player changing the rules because they are losing the game, and typically manifests when a person does not want to admit you have a point, or concede theirs (as Dr Bo put it, "failure to be an honest interlocutor").

The quick answer is that if everything is subject, then is the statement "everything is subjective."  It is internally contradictory.  I could write a lot more on this subject, but I have to run at the moment.