Adorkable: The Russian is an evil example. Affectionate towards his allies/friends, wants to get "lots of Levis and CD's" with a million dollar reward, and is openly a fan of GOOD superheroes (Mighty Thor Good Communist, with that Big Hammer of his!).
Frank is usually treated as an absolute moralist who just happens to be fighting a never ending war on crime. Other interpretations imply he's just as bad as the monsters he fights and is little different from a serial killer. The only thing that stays the same is the reason why he became the Punisher.
Punisher: Year One has yet another take on Frank's character - he goes vigilante only after all other methods of bringing the mobsters who killed his family to justice fail spectacularly (the police are ineffective and corrupt; an attempt to do an independent investigation leads only to the death of the reporter whom Frank briefly befriended).
Punisher: Born from MAX imprint has a different take on WHY he became the Punisher. Frank fell in love with killing and made a deal with an unknown supernatural entity (or just plainly went mad, you choose) to fight a war that would last forever. He wasn't informed until later that the price would be his family, at which point the supernatural entity makes Frank forget that they ever made the deal.
Do Not Do This Cool Thing: Punisher comics are often accused of tacitly endorsing murderous vigilantism, especially since the mid-2010's as various social and political factors made people increasingly uncomfortable with the idea of an armed man roaming the streets as self-appointed Judge, Jury, and Executioner. Marvel have tried to defend themselves from this by claiming that Frank is depicted as a deeply flawed man who should not be admired or emulated, with the character stating In-Universe that nobody should follow his example and rejects anyone inspired by him. However, even in his most morally dubious incarnations, Frank is a badass Action Hero with a Sympathetic P.O.V. killing the most vile people imaginable, so it's a little hard to argue that readers aren't supposed to think he's awesome and wish they could be like him.
In The Punisher: Purgatory, Frank was turned from a mafia-hunting anti-hero into a supernatural force of vengeance on a path to redemption to be able to rejoin his family in heaven. This was undone with a Handwave by later writer Garth Ennis in his "Welcome Back, Frank" storyline. Despite the handwave, a subtle nod to the Heavenly Enforcer period appeared in a volume two Thunderbolts issue, after Deadpool acquires an angel feather while in Mephisto's hell it heals Punisher from fatal injuries he'd received while he was still on Earth. Deadpool asks how someone so non-Heavenly as Punisher could possibly have anyone in Heaven interested in him (to paraphrase) only for the Punisher to blow him off with a "don't want to talk about it".
Likewise Franken-Castle will be remembered fondly as a brief period of lunacy in Frank's life. There's no way the powers that be intended for a magic/SCIENCE half-robot Frankenstein's Monster Punisher to be a new cutting edge status quo. Even in the Heroic Age. These days Franken-Castle exists only as an AU Punisher from a Monster Mash universe. He is part of the Avengers there (which also include monster forms of other heroes who had them in the mainstream 'verse in the past, like werewolf Captain America - see below - and vampire Wolverine). They only appeared briefly to tangle with the regular New Avengers and were offed unceremoniously.
John Ostrander's oft-forgotten run had Frank become the head of a Mafia family before things turned into an X-Men story that happened to star the Punisher, with Frank being roped into helping X-Cutioner and S.H.I.E.L.D. rescue an activist from the latest incarnation of the Mutant Liberation Front (here a false flag group backed by Humanity's Last Stand). The final issue had Frank incur amnesia, leading to the aforementioned Purgatory. Ostrander killing off some well-liked characters doesn't help.
Frank Castle inexplicably joining Hydra in Secret Empire has been met with nothing but scorn by Punisher fans. Some believe that Nick Spencer only did this to anger Punisher fans.
Frank and Microchip would come off like an old married couple sometimes. In some issues, Micro would even use the word "bubbelah" (Yiddish for "darling") when talking to Frank.
Punisher: Up for thirty-six hours watching a crack lab, and Micro calls me with a grocery list on the way home. Sometimes having Microchip is like having a wife.
The most overt instance of this probably the first issue of War Zone, where Frank and Micro bickering over Micro sneaking off to see a therapist comes off like a couple cracking up due to one member uncovering another's infidelity.
Microchip: ¢%$#! I don't need this, I'm going out.
Punisher: Where? I might need you later.
Microchip: Just out. *¢%, we're not married, Frank.
Franken-Castle's fight with Daken was filled with homoerotic banter.
Gerry Conway has stated he's uncomfortable with how popular the Punisher is with members of the American military and law-enforcement, as he is very anti-war and anti-authoritarian as well as missing the point of a vigilante. He intended for the Punisher to be a tragic anti-hero with ambiguous and sympathetic traits but as primarily an example of the failure of America's military and legal systems:
Gerry Conway: "The vigilante anti-hero is fundamentally a critique of the justice system, an example of social failure, so when cops put Punisher skulls on their cars or members of the military wear Punisher skull patches, they're basically sides with an enemy of the system. They are embracing an outlaw mentality. Whether you think the Punisher is justified or not, whether you admire his code of ethics, he is an outlaw. He is a criminal. Police should not be embracing a criminal as their symbol...My point of view is, the Punisher is an anti-hero, someone we might root for while remembering he's also an outlaw and criminal. If an officer of the law, representing the justice system puts a criminal's symbol on his police car, or shares challenge coins honoring a criminal he or she is making a very ill-advised statement about their understanding of the law."
An in-universe version happens when Frank meets a group of fanboy cops who actually did put his skull on their car. Frank rips it up, and tells them that nobody should be like him and if they want a role model, Captain America is always around (see Harsher in Hindsight above).
Discussion amongst comic professionals and fandom is also starting to highlight the use of the Punisher skull amongst some factions of politically violent militias in the United States, even though Frank himself has never expressed sympathies with any of those ideologies. Garth Ennis, who wrote the very violent The Punisher MAX amongst other Punisher series, condemned the misuse of the skull quite harshly, but there have been requests aimed at Marvel to redesign the skull, reaffirm Frank's condemnation of militia violence, or just retire the Punisher entirely. Marvel, however, may keep using the character and the logo nonetheless.
Never Live It Down: Seeing that the writers never bothered to fix Castles background after Secret Empire. In-Universe, Frank will always be remembered as someone who joined Hydra, and him trying to atone for what he did by hunting down every Hydra agents and yet Heroes keep undermining him from doing so.
Sequel Displacement: While Frank was one of Marvel's cash cows back in the early nineties (alongside Spider-Man and Wolverine), many of his modern fans know nothing about his stories before Ennis. Sadly this means that a lot of his gentler characterizations are overlooked, as well as the fact that he actually did have something of a supporting cast.
Carlos Cruz who has some interesting skill and backstory and is molded by Microchip to try and replace Frank, is in the series for a disappointingly short time and only has one significant fight with Frank, when their feud could have been dragged out for longer and/or eventually made way to an easy truce.
Rosalie Carbone, during the Jon Ostrander run. A somewhat long-running, decently layered villainess who still wants Frank dead at a time when he's in a position of unprecedented strength and is also revealed to have some history with the Punisher's erstwhile allies the Geraci's, this could have been expanded as a notable subplot but is only in one issue of that arc.
Stuart Clarke's job as Microchip's replacement and a former super villain feels like it could have been explored for a lot longer and given both him and the Punisher positive development under the right circumstances.
What Do You Mean, It's Not Political?: Punisher fights against common and grounded criminals (kidnappers, narcos, drug dealers, mafias, rapists, serial killers, and the like) more often than against supervillains with weird superpowers. The The Punisher MAX series is in fact strictly set in such circumstances, with other superheroes and supervillains being virtually absent. Punisher often kills those criminals, with no remorse and extreme prejudice. This made him popular among people who support hard-line policies against crime, and even far-right extremists. The comic, however, is devoid of such political content: the actions of Punisher are shown as just the actions of Punisher alone, a single vigilante, and never try to address the issue of the way the government or society at large should deal with crime. People from the government, police, and military that appear in his stories are usually shown as either incompetent or corrupt.
Complete Monster: General Kreigkopf is the leader of the Russian Syndicate in the second portion of the game. An associate of the Gnucci Family, Kreigkopf would supply them with drugs and money from the harbor. When Castle attacks the harbor, he discovers Kreigkopf has been smuggling in weapons, including a tank and a nuke. Frank discovers that Kreigkopf is also part of a Human Trafficking ring. Kreigkopf sends the Russian to attack Castle at his apartment. When Castle and Fury invade Grand Nixon Island, he prepares to launch the nuke at New York City, and manages to get it to the point it's supposedly irreversible before Frank kills him.