Satirical depictions of politicians are almost inevitably popular with their targets (with the notable exception of Steve Bell's take on former British Prime MinisterJohn Major.) Often, they will contact the cartoonist, or the paper it was published in, to ask for a copy or the original, probably thinking it's better if people are making fun of them than just ignoring them. Ralph Steadman declared he would only depict politician's arses to prevent this.
Super-Mac by Victor Weisz, a parody of Harold Macmillan, was especially so. Maybe he shouldn't have compared him to a superhero.
Astérix: Sometimes used by European far-right politicians or supporters to promote the idea that ancient Europe was far better because there were no immigrants like today. It doesn't occur to them that "Asterix" is more historical fiction than anything else and that Asterix and his friends always get along fine with other nationalities. Even the Romans aren't always depicted as villains.
In the infamous Chick Tracts, readers are supposed to agree with everything the protagonists say, but there is a significant "fandom" that finds the over-the-top nature unintentionally hilarious. In addition, on first reading them, many people assume that they are intended as a parody. They are serious. The sheer number of times he has Straw Secularists/Liberals (especially in schools), such as the dystopia in "Last Generation" which has the security and language of Oceania, the religious politics of Left Behind, and the social politics of Straw Liberal states, with a touch of "concentration camps" for parents who discipline their children — it makes it difficult for one to accept them as serious arguments unless one realizes that there are more extreme people out there.
This happened to R. Crumba lot — most notably with his iconic "Keep On Truckin'" character/pose, which was adopted by many rock-loving hippies as their "mascot," as it were. The truth was, Crumb was making fun of rock music lovers, who in his eyes were doing "The Dance of Cultural Death" (as he put it). He even explained it in a comic in The R. Crumb Coffee Table Art Book and told his (probably now disillusioned) hippie fans: "KEEP ON TRUCKIN', SCHMUCKS!". (This was followed by Mr. Natural remarking: "Don't forget, Bob, that it was the compassion, the loving forgiveness, that they found so appealing in your cartoons, that made you so popular, that got you laid, that earned you a living. Keep it in mind!")
The Dark Knight Returns has an in-universe example: The Sons of the Batman, a group of vigilantes inspired by Batman using incredibly violent methods against mostly petty criminals (ie, stopping a three card monte game with napalm, cutting off the arms of a shoplifternote Not to mention the fingers of the poor clerk, for not defending the store.). Needless to say, when Batman finally meets them, he sets them straight.
This goes for many popular monstrous characters, there is a difference between enjoying the character's appearance in the story (which, one must imagine, the creator wants you to do unless stated otherwise), and the kinds of interactions they bring, and seeing their crimes and psychopathy as something to be cheered on and supported or thinking the villain might be a cool dude to know, which is the main idea of this trope.
A better example from the pages of Batman might be Harley Quinn. Although she is the girlfriend and accomplice of The Joker, and is often shown to be almost as Axe Crazy as he is, fans often seem to forgive her actions, hold her up as something of a heroic or anti-heroic figure, and she is often a Karma Houdini in the actual stories.
Batman in general isn't necessarily immune to this. Mark Waid's Justice League story Tower of Babel was designed to criticize the character's prep time paranoia tendencies by showing that he'd secretly been thinking up ways to kill or incapacitate his Justice League allies for years, only to have them fall into the wrong hands, thus placing the entire world in jeopardy, but unfortunately all some fans came away with was "BATMAN'S THE SMARTEST, MOST BAD ASS HERO EVER!!!"
Judge Dredd. You'd be surprised how many people find the idea of the Judge system appealing and miss the strip's satire altogether.
Kingdom Come: Some people read it just because they like the Antiheroes. This is missing the fact that Kingdom Come was written as a criticism of that kind of character. Others miss the idea that a big part of the story is that Superman and the new League trying to bring about world peace works horribly and ends up getting everyone nuked, and wholeheartedly support/condemn them as Silver Age nostalgia.
Some of that has to do with the concepts that Waid and Ross came up with being popular enough with writers that they were madecanon. A few characters like Irey West, Jakeem Thunder and the female Judomaster ended up crossing over into the DCU, while Cyborg temporarily got his golden skin and Roy Harper became Red Arrow. Seeing as how those characters were generally not shown to be outright asses though, it's somewhat understandable.
It got to the point that Magog, who existed exclusively as a self-righteous Take That aimed at 90's antiheroes (Cable in particularly), was given his own book that played his over-the-top attempts at badassery straight. The title itself was cancelled pretty quickly and Magog ended up being killed off shortly after it ended.
Magog even got Misaimed Fandom from his creators. Waid and Ross tried to design his costume to include everything they hated about 1990s costumes, but ended up kinda liking it. The character also gets a clear shot at redemption.
Lobo started as a generic mercenary before being retooled by creator Keith Giffen as a parody of eighties "grim and gritty" heroes like Wolverine and The Punisher in a series of mini-series books. Needless to say, Lobo became a big hit with fans who took the satire at face value.
This happens with a lot of "satire" characters where the author "exaggerates" them just by taking all the elements that people seem to like in other shows and lumping them together without actually exaggerating anything. We've seen this in reverse with films like Sucker Punch, intended to "parody" exploitation literature but garnering reactions as if they were genuine because, well, the creators forgot the part where they make the thing they're parodying more ridiculous or extreme than the source material.
German comic Nick Knatterton was made as this, since author Manfred Schmidt considered comics a primitive art form. The fans took it straight and liked it.
Superdupont is a French comic parodying the superhero genre and a satire of French jingoism. The titular character is an over-the-top stereotypical Frenchman with Superman-like powers (which he loses when he hears the French anthem played in reverse) and battles "Anti-France", a shadowy group of people who all speak with a mix of all foreign accents at once and target French core values - such as replacing French wine with Italian wine and mass-producing berets made in China. The French extreme right-wing nationalist party took Superdupont as their icon, which caused the authors of the comic to put it on hiatus for a few years.
Jhonen Vasquez repeatedly takes pages out of his Johnny the Homicidal Maniac and Squee series to Take That to various people he feels are enjoying his comic for the wrong reasons. One extended story in Johnny the Homicidal Maniac is about a serial-killing fanboy of Johnny's. Since Johnny is a character who goes around murdering the most annoying people in the typical Vasquez Crapsack World, it's not hard to see why some people might get the wrong idea.
Goths seem to treat Jhonen as their king, despite him constantly insulting them and his own hatred of the association. With that said, he doesn't necessarily hate Goths, but he doesn't care for catering specifically to them.
V from V for Vendetta, to the point where the live action adaptation made it so that he was obviously meant to be the hero. V is a fanatical terrorist whose main motives are revenge and his methods include physical and psychological torture (of both enemies and allies), bombing of public monuments, and brutal murder. An argument can be made for a case of A Lighter Shade of Grey, given that V is also a charming and charismatic Noble Demon and his enemies are a brutal, genocidal and largely irredeemable fascist regime, but V was intended to be a lot more ambiguous than many ultimately view him as being.
V for Vendetta, particularly the movie, spread the misconception that Guy Fawkes Day honors Guy Fawkes, the plucky rebel, instead of celebrating the fact that England narrowly averted a terrorist attack on the capital. It's like thinking September 11th honors Osama Bin Laden. The holiday also has anti-Catholic overtones (Fawkes was a convert to Catholicism), which makes it particularly ironic in the film when V slays a pedophilic Catholic priest.
In Watchmen, a '80s superhero deconstruction, Alan Moore heavily based the character Rorschach on Steve Ditko's Objectivist superheroes, specifically The Question and Mr. A. However, Moore had no affinity for their ideology, calling Mr. A "an absolute insane fascist" and Objectivism "laughable," and he wrote Rorschach as his own take on what an Objectivist hero would probably be like, a short, ugly, murderous sociopath. Despite this, readers saw Rorschach's uncompromising persona as endearing, and he became the most popular character of a landmark comic series. Additionally, as pointed out on the Unbuilt Trope page, Rorschach and the Comedian were intended to deconstruct the Nineties Anti-Hero, and ended up popularizing it instead. Apparently, the series's beginning with the horrific death of the Comedian and ending with the even more horrific death of Rorschach wasn't enough to make people realize that these were not admirable characters.