Batman has several foes who, despite their gimmicks, choose to go by their real names. Hugo Strange, Cornelius Stirk, Victor Zsasz, Philo Zeiss, David Cain and James Gordon Jr. Mister Freeze might also count, since his legal name is Victor Fries (pronounced exactly like "freeze"), so spoken out loud, his code name is just his real one.
Discussed in Adventures of Superman #574, that opens with Superman fighting a guy in a power suit who identifies himself as "Gabriel Von Daniken." Superman tells him that it's the worst name for a super villain and he's ever heard, and Gabriel replies "You mean just because I build a containment suit and try to poison the water supply I have to give myself a stupid alias? Get a grip, Superman. I'm thirty-five years old!" A bystander watching the battle remarks that he has a point. After all, Bonnie and Clyde never had code names, and they were criminals anyway. He says it's actually kind of cool for villains to introduce themselves by their real name, instead of calling themselves "Toxic-man" or "Pollutus".
Supergirl spent the first half of Supergirl (2005) and the entirety of Supergirl (2011) going by her real name -Kara Zor-El-, without bothering with a secret identity.
Doom Patrol has Scott Fischer. Apparently, they gave him the codename Blaze but he never got around to actually using it.
During the Mike Grell run, Green Arrow and Black Canary largely dropped the costumes and codenames. Later, Canary would all but abandon a civilian life, and Green Arrow's status would change Depending on the Writer.
Mary Marvel is a borderline case, as Mary is her real first name. For a while after Captain Marvel became Shazam!, none of the rest of the Shazamily really had a codename, then Mary was The New Champion of Shazam! for about ten minutes, then she went back to Mary Marvel.
Mal Duncan, who later went through an ever changing series of codenames: Guardian, Hornblower, Herald, and now Vox.
Danny Chase did not have a codename, nor any form of costume. He temporarily came up with the "Phantasm" identity while pretending he was dead shortly before being written out of the series.
Raven, who's from a magical other dimension, operates by her (singular) real name. As of late she's used the civilian identity "Rachel Roth" , but that name is the made up one.
In Young Justice (2019) Tim Drake came across an evil alternate that was just going by his own name in costume. Tim, who has a long history of trouble coming up with aliases for himself, decided that if his alternate couldn't come up with a name he probably wouldn't either and started going by Drake. By the end of the series, he'd gone back to Robin.
After the various Crisis Crossover-induced Continuity Snarls in her back-story, the first Wonder Girl called herself Troia for a time, but eventually settled on just using her civilian name of Donna Troy.
Many of Wonder Woman's villains operate without code names even if they do use gimmicks, including Mona Menise, Circe, Veronica Cale, Paula von Gunther (who has spent more time as a reformed villain and loyal ally to Diana than she has as her opponent), and Zara.
A similar borderline case to Mary Marvel is Johnny Chambers/Johnny Quick and his daughter, Jesse Quick, although Jesse Chambers later adopted her mother's identity as the second Liberty Belle.
Western Animation
Justice League: Wonder Woman is never "officially" given the name Wonder Woman. She is addressed as such on only extremely rare occasions and only by people who are not very close with her (e.g., a bouncer at a nightclub, the obnoxious host of a talk show actively slandering the League, and Lex Luthor in the midst of battle); in every other situation she is simply "Diana". J'onn J'onzz is addressed as the Martian Manhunter only once in the entire series, in the briefing for Task Force X in the second season of Unlimited. These two characters do not have a Secret Identity or any life outside heroics, so they have no need for code names or hiding. Wonder Woman was shown to be an ambassador in one episode, so a secret identity would be all kinds of impossible.
Young Justice (2010): Artemis Crock uses the superhero name of...Artemis. Somewhat Justified because her work on the Team is covert ops, so she's not well-known publicly, and because her name fits her gimmick rather well, assuming that you're familiar with Classical Mythology. This leads to an important bit of foreshadowing in "Targets" when Cheshire refers to her as "Ar—chery girl." This hints at the fact that Cheshire knows Artemis's real name, but not her codename. The reason, of course, being that they are sisters.
