I believe it’s real, but frequently misapplied to situations that are better called something else. Is it “cancel culture”, for example, to fire someone for any reason other than inability to do their job? Does a company have a right to protect its reputation and enforce its own rules (and fire people if they violate them, even if what they’ve done is not illegal)? Is it "cancel culture" to exercise economic freedom and boycott a business whose practices you don't support? A lot of what gets called “cancel culture” is just the equivalent of an audience booing someone on stage. It isn’t “cancel culture” to be poorly received or criticized. It’s just as “canceling” to say we can’t react negatively to someone or their content.
It’s not new for people to be shunned or blacklisted, but I think what “cancel culture” refers to is the relatively recent social media-inspired phenomenon of a public person doing or saying something stupid or “problematic” and then being removed from their job or having their reputation ruined (and often very quickly without much introspection). I’ve always been someone who can separate art from the artist. I’m not going to stop listening to Wagner because he was an anti-Semite or stop watching Woody Allen films because of the allegations against him. I think what alarms people about cancel culture is that often it involves allegations only (no proof of any wrongdoing), it involves the dredging up of old content that the person has since disavowed and thus perpetuates the idea that people cannot escape their past or redeem themselves, and the swiftness and ruthlessness with which it happens.
I also believe that referring to it as a "culture" is appropriate given the recent instances of "self-cancellation" where entities (like the Dr. Seuss estate) censor themselves or their own work in anticipation of a "cancelling mob" that may never have actually been gunning for them--nonetheless the culture is such that this is even a concern. But as others have pointed out, the idea that this exists only on the left is patently false; right-wing "cancellation" is attested from the past (particularly on the part of religious conservatives going after elements of culture they view to be debased and sacrilege) and occurs now.
"Cancel culture" is intellectual laziness. It's removing something before you can engage with it rather than engaging with it honestly and fairly.