In the Batgirl series, Barbara Gordon, the originalBatgirl and at that point the tech-savvy Oracle (although now she's Batgirl again), refuses to call Stephanie Brown by the name "Batgirl" after she takes over the persona. This is for a variety of personal and professional reasons, and she always refers to Stephanie by her proper name or with various pronouns, always avoiding the actual word "Batgirl." When she finally does use the term Stephanie recognizes that this signifies her acceptance and approval, and it gives her the motivation to persevere when fighting the Scarecrow.
In one Bloom County storyline, Opus becomes a garbageman but insists upon calling himself a "waste-management artisan". When Milo calls Opus out on how dumb that sounds, he asks, "What did Ronald Reagan call those weapons shipments to Iran?" Milo answers "'Goodwill gifts'", to which Opus replies "I'm a waste-management artist, thank you very much."
The Buffy the Vampire Slayer: Season 8 comics have Dawn cursed to turn into three things. She would like you to know that the second one was not a centaur, it was a centaurette, which is, as she defined it, a girl centaur.
"Calvin the BOLD demands he be addressed properly."
"You seem to have mistaken me for some mild mannered youth! I am STUPENDOUS MAN!"
Death's Head, of the Marvel universe (and occasionally appearing in the Transformers and Doctor Who comics when Marvel had the license), is a freelance peacekeeping agent. Some call him bounty hunter, but never twice, yes?
Ghost Rider antagonist and Legacy Character "The All-New Orb" literally always refers to himself with that full phrase. Others just call him "Orb" or "The Orb", but he's been very clear what he'd prefer to be known as.
In the Hellboy comics, the Nazi head-in-a-jar mad scientist Herman Von Klempt gets very upset if you call his ape a monkey.
In MAD, this is often used for humor. For example, one article described wealthy people's activities less bluntly than poor peoples' ("When you're poor, you vomit. When you're rich... you succumb to a sudden attack of nausea.") Another article tells people to think of others' behavior in worse terms than their own- others are pushy, while you are assertive.
Max from Sam & Max: Freelance Police dislikes being called a rabbit, and instead prefers the term "Lagomorph" or "Hyperkenetic Rabbity Thing." He dislikes being called a "malefactor" too. But for really insistent terminology, just use a really weird or long word around him. He hates that even more.
The Sandman brings us Fun-Land. "Not FUN! FUN-LAND!" This being The Sandman, it's actually pretty terrifying: the guy in question is a child-murdering sociopath hiding behind the mask of an amiable, slightly goofy fat guy who stalks his "prey" in amusement parks.
Spider-Man has been known to correct villains who call him an insect, pointing out that spiders are technically arachnids.