Claiming statements as opinions

Essentially, it is someone claiming that their disagreement with an established fact is their opinion, then there is an implied claim that all opinions are equally valid.

Can one have an "opinion" on whether or not to accept an established fact? I guess. This might get into the difference between an ontological claim and an epistemological claim. When someone expresses an opinion, I would think they are not claiming a fact about reality, but saying what they think or believe about reality. Given that, I would say having an opinion about a fact is more of a problem of ignorance in that if they had the knowledge, they would simply accept the fact.

I do think a case can be made for a false equivalence . Again, the implication is "I have an opinion and you have an opinion, so both are equally valid," but one "opinion" is really a fact.

Several issues come to mind:

The problem with “I’m entitled to my opinion” is that it’s used to shelter beliefs that should have been abandoned. It becomes shorthand for “I can say or think whatever I like” – and by extension, continuing to argue is somehow disrespectful. And this attitude feeds into the false equivalence between experts and non-experts that is an increasingly pernicious feature of our public discourse."  ( See Patrick Stokes: The Ethics of Bravery)

Not only is "I'm entitled to my opinion" an old trope but like the planet Pluto itself it has been downgraded and reclassified as a "thought-termination cliché" ( See Robert Jay Lifton: Thought Reform and the Psychology of Totalism). 

The academic response to the weak and childish argument of  I'm entitled to my opinion is, no, you're not, you're only entitled to what you can support. 

As for the IAU, Pluto is now classified as a "dwarf planet" as it doesn't meet the  consensus criteria for a full-sized planet:

It is in orbit around the Sun.
It has sufficient mass to assume hydrostatic equilibrium (a nearly round shape).
It has “cleared the neighborhood” around its orbit.

Since Pluto only meets the first two out of three it is classified as a "dwarf planet". So, the claim that it is still a planet is not entirely wrong, it's just not scientifically up-to-date. 

I think the argument here is:
"Something is true because that's what I think (i.e that's my opinion) and you can't criticize me of being wrong since it's my opinion."
First I would say that it is his opinion but it is also a SELF-DELUSION. I don't know if it there's a name for the aforementioned argument [so typically this is not an answer to your question - maybe I should write this as a comment(?) I dunno] but I would suggest reductio ad absurdum as a response, for example
"Humans have five legs. It's just my opinion"

Astronomers in our midst would probably address the "Is Pluto a planet?" by offering accepted definitions of "planet" and then going on to compare Pluto's characteristics with the definitions.  If all of the characteristics are met by Pluto, then it stays in the club; if Pluto doesn't match the definition, then it's out.  We already know the recent results of that exercise.

So, let's look at "just my opinion" in the same way.  I'm not sure of the absolute authority about opinions; however, some examples are available.  Merriam-Webster seems to think an opinion is "a belief based on experience and on certain facts but not amounting to sure knowledge".  Various news sources consider opinion to be "thoughts and analysis on certain issues".  Oxford Languages sees an opinion as "a view or judgment formed about something, not necessarily based on fact or knowledge".   Way back, the folks from "Laugh In" would probably have suggested that we "look it up in our Funk and Wagnalls" – if we did, we'd find it to mean "A confident belief without full certainty".  Even Wikipedia gets into the action with "a judgement, viewpoint, or statement that is not conclusive, rather than facts".

Two common threads appear among these definitions of "opinion": (1) it's a belief or a judgement, and (2) it's not based on fact and not certain.

So, are we entitled to our opinions?  Certainly!  It becomes a non sequitur when we assume that an opinion can be used as a fact – that's when logic breaks down.

As an aside, I think it can be a good thing when "just my opinion" comes into play, even if it goes against accepted positions.  In keeping with our astronomical theme, way back when wasn't the "accepted fact" the notion that all things revolved around the Earth?  At that time, wasn't there someone who said something like "It's just my opinion, but I don't think the Earth is the centre of things ... and I'm going to work on demonstrating that my opinion is correct."?

Just saying "this is my opinion ... even if it goes against accepted facts" isn't in itself a fallacy; treating it as a fact and developing further conclusions claiming the results as fact is where the fallacy rests.