The original concept of the Fake Right comes from Hofstadter, who borrowed it from Adorno. He used it to refer to so-called conservatives who refused to compromise politically with the loyal opposition, the so-called liberals.
I would say that “Right” and “Left” are ideological labels dating to the old days when parties were ideological, as in the French Revolution. Nowadays, most people think ideology is nonsense, because the ideological Big Gorillas—communist tankies and absolute monarchists, along with their religious fellow-travelers—look stupid from a 21st-century US perspective.
Since US political parties change policy positions every few years, and political animals spend all their time mocking and insulting ideological purists, it seems like ideological claims are now just empty rhetoric. So, I would say that most professional politicians or professional activists claiming to be Right or Left are fake. The ones who actually believe what they are saying are always cut down and ignored by the operational folks on their own side, who are embarrassed by the naive believers.
That’s why we have the Fake Left who hate ordinary workers and the Fake Right who hate tradition. Meanwhile, the Fake Left and the Fake Right are in complete agreement on the important things, like how to spend government money, how to keep corporate donors profitable, and how to control their own citizens.
I’m not even getting into “woke left” and “alt right.” Those are non-ideological shock troops intended to keep each group of animals in their own pen, bleating loudly at the animals in the other pen.
The livestock analogy for politics nicely illustrates why we seem to have 50/50 splits in the electorate. When the “two sides” have the same motivations, their steady-state condition has to be a perfect balance. They both control their stock by acting big and by exaggerating how big the other group looks, so they always appear to be in balance. Meanwhile, surveys of the population show that the image of both sides is fake, since together they might be lucky to represent a majority of the population. It would be more accurate to represent political animals on one side and normal people on the other side.
But, when you add in a bunch of cattle drivers with electric prods (the woke left and the alt right) forcing everyone outside the pens to “pick a side”, then you get the “polarization of society” that fake journalists whine about. (If they are whining about polarization while openly calling for the imprisonment of one side, then they are fake journalists.)