Yes, the term "political correctness" originated on the state left -- but it originated as a slam. Non-Stalinist socialists used it to disparagingly describe Stalinist communists who changed their stated views to conform to whatever line was prescribed in Moscow.
The obvious example of "politically correct" back then would have been the Stalinist who agitated against US intervention in World War II while Germany and Russia were having their "non-aggression pact" honeymoon in Poland, then turned on a dime the day after Hitler launched his attack on the Soviet Union and started demanding that same US intervention he'd complained about earlier.
Even as late as the 1970s, it remained a sort of inside joke on the state left, basically used to mock people who thought that their organizations' political lines should determine their thoughts rather than the other way around.
But there existed a fatal crack in state-leftist political thought that made the term ripe for an exploitative right-wing backlash. That crack was the increasing adoption of post-modern "critical theory" thinking, an abscess of irrationality that has, over the last 50 years or so, infected and discredited large segments of the left (including, lately and much to my dismay, the libertarian left) such that, not unlike Orwell's Newspeak creators or the anonymous Bush 43 aide in Ron Suskind's telling, its adherents believe they can (and should) control reality by seizing control of narrative (which is to say, seizing control of speech). Which leads us to today's authoritarian idiocy like claims of "microaggression," demands for "trigger warnings" and "safe spaces," and attempts to shut down competing views via "no platform" demands instead of engaging and defeating those competing views in debate.
Those implications were still in their infancy when the right enthusiastically counter-adopted the term "political correctness" to describe any non-right-wing political view, and evolving to now when it's used to dismiss -- without the necessity of actually making a sound argument -- any criticism of any right-wing person's political views.
So here we are at the current state of both evolutions.
Although the non-libertarian left eschews use of the term "political correctness" these days, the pomo/CT line has obviously emerged triumphant (at least on college campuses and among the "youth" wing). If you disagree with a leftist claim or demand, you're not just incorrect and your point of view is not just needful of logical countering -- you're evil and your point of view constitutes aggression. It is therefore justifiable self-defense to violently suppress expression of that point of view, or even objective coverage of your own point of view that might be embarrassing ("I need some muscle over here").
On the right, use of "political correctness" has evolved from a dismissive insult versus leftists to a sort of "get out of opprobrium/ridicule free" card for rightists like Donald Trump ("Every once in a while, I like to be politically incorrect") and Hans-Hermann Hoppe (he "pays no mind to political correctness," see?).
Both versions/approaches serve the same purpose. That purpose is to short-circuit substantive discussion so that one's claims can be put over without the need to actually justify them. Although to give the right its due, they haven't pushed it in the direction of direct forcible censorship in quite the same way the state left has.