Characters in entertainment mediums can vary massively in their importance to narrative. They can be present for comedic relief, the primary protagonist for the viewer to identify with, or they can drive the narrative in its entirety. While I believe characters are almost vital for a narrative to occur, I don't believe that characters are inseparable from their stories, or monogatari.
As discussed in the Condry reading, characters can be incredibly important to the stories that they're in, and in many ways, their interactions can drive the story. The introduction of a new character can be important for the plot of an episode, or an entire series or story can be constructed around single characters. For example: Zenmai Zamurai, an unimplemented cast member of the Japanese shorts "Dekoboko Friends", had an entire series crafted around his existence. While not as extreme as the Zamurai example, the characters of Dragon Ball Z exemplify the importance that characters can have to the narrative of both anime and manga. More so than the premise and the setting, the Dragon Ball Z television show's events are almost entirely character driven. Seasons, and multi-season events are often based off the introduction of new characters, who are often antagonists. As can be seen in this article nearly every season of the Dragon Ball Z series is named after the characters or antagonists that are introduced during the season, and the story is most definitely driven by the existence and actions of the antagonists. The seasons/ story lines begin with the introduction of a villain, and end with the defeat of the villain. When consumers engage with these characters in a medium outside of the original manga or TV series, they're engaging with characters that are not just a part of a grand narrative, they are the grand narrative. Freiza, Cell, Buu, all purchasable action figures, and all embodiment of extended story arcs, not just glimpses of the Dragon Ball Z storyline. While the characters of DragonBall Z are entirely vital to the narrative of the show, if donjinshi creators so please, the characters can absolutely exist outside of the story that they're created in. No characters are safe from transmedia, even if the medium is not cannon.
While I believe that there are no monogatari characters that do exist, due to the existence of parody manga and individual narrative developments, there are characters which are awfully close to being unable to exist outside of their story. Super Mario, for example, is a cannibal of game titles. Mario's existence within a game, especially when he is a playable character, almost guarantees that he will not only be a marquee part of the game, but will have his name in the title. See this list of titles Mario is featured in. After 1986, nearly every game that features Mario as a playable character, has his name in the title. Mario's existence within a game bends the narrative around him, and designates the game as a part of Mario's story. In games like Mario Kart, Mario Tennis, and Mario Party, where Mario's presence as a playable character is no more important than any of the others available, his existence will induce Nintendo to brand the game under his name, and potentially utilize him as the game's narrator. While this is done to increase sales and interest in the game title, it is an unfair individualization of game narratives by one character. The utilization of popular characters in exotic medium is exactly what separates them from their story. Super Mario's appearance on a lunchbox hardly corresponds and reflects his battles with Bowser.