A Super Hero who does not actually have a hero name, despite maintaining a generally low-profile Secret Identity. These heroes are simply called by their "real" names in the heat of battle.
This is not a character using a family or given name that is snazzy (or not) as his or her heroic name (for example, former surgeon Stephen Strange continuing to go by "Doctor Strange"). Compare Steven Ulysses Perhero, where the Meaningful Name has something to do with the character's powers.
Compare Overt Operative, a secret agent who never bothers keeping low.
Jean Grey went without a codename for some time in comics, and has been codenameless in most screen adaptations (1990s series, X-Men: Evolution, movies). This is because, by the time she came Back from the Dead in The Bronze Age of Comic Books, code names for adult characters ending in "Girl" had gone out of style, so she couldn't use "Marvel Girl" anymore, and "Phoenix" was attached to a different concept.
Ironically, given the opening quote, the teenager legally known as Jubilation Lee used the codename "Jubilee" for many years; this could be argued to be about as much a codename as Peter Wisdom calling himself Mr. Wisdom. She went by "Wondra" for a while, though.
On the other hand, orphaned Jubilee did not have a family or life outside the X-Men, while Jean Grey still had parents and a sister with a family of her own.
It was intended that Zaladane have powers because she is related to Lorna Dane. Claremont forgot, however, that Lorna is adopted. (Source:X-Men danglers list)
Lorna Dane wasn't called Polaris until Chris Claremont took over. She operated as just Lorna for some time.
And the name Polaris was first given to her by a mind-controlling villain (although Classic X-Men incorrectly showed the name earlier).
This was also lampshaded in a commercial for X-Men: Evolution, where the other members were introduced by their names and code names (for example, "Scott Summers is Cyclops"), but hers was simply, "Jean Grey is... Jean Grey".
Heck, they discuss it in-show. "How come everyone has a codename but Rogue is just Rogue?" "Same way Jean is just Jean."
Emma Frost, since joining the X-Men, dropped her codename "White Queen". Justified in that "White Queen" isn't just a name, it's a rank in the Inner Circle of the Hellfire Club. Which she left to join the X-Men.
Cecilia Reyes never chose a codename because she didn't want to be a superhero. But she had the X-Men Bumblebee Suit and she was considered one of them after she was outed as a mutant.
Dani Moonstar, formerly Mirage and Psyche, eventually dropped her codename and just went by "Moonstar". Admittedly, people who don't know her secret identity might well assume that Moonstar is her codename. Also, her "civilian" identity is an agent of SHIELD, rendering the "secret identity" somewhat moot.
Let's face it: X-Women tend to lose or not have code names (or have codenames that are for all intents and purposes their real name, like Rogue and X-23), possibly because their creators really like the real names they really like (Pryde, Frost, Grey) or introducing them as civilians who eventually chose half-hearted codenames that never really stick. Storm may be the only major female X-Man who hasn't operated for a significant length of time without using her codename (Psylocke has used hers pretty consistently since she got it, but started out as "Captain Britain's sister Betsy" and briefly "Captain Britain").
Bucky, Captain America's Golden Agesidekick and current Captain America, was actually called Bucky Barnes, as a nickname based on his middle name, Buchanan. What's different, and very odd, about this is that "Bucky" was treated as a code name, and nobody knew that Bucky Barnes was the same person as Cap's sidekick Bucky.
This sort of thing happened a lot during The Golden Age of Comic Books. This includes sidekicks Roy the Super-Boy [who worked with The Wizard], Mickey Mathews [The Deacon], Tommy the Amazing Kid [Amazing Man], Rusty [Flagman], Sandy the Golden Boy [Sandman] and Mickey [American Crusader].
Speaking of Roy the Super-Boy, his Secret Public Identity was lampshaded in one story, where, when calling his friend's mother, he identified himself as Roy. When she asked him if he was her son's friend, he insisted that no, he's Roy the Super-Boy. For some reason, she buys it without question.
Black Terror's sidekick, Tim Ronald, is something of a twist on this trope. He was initially known as Tim in his superhero identity, but around the late 1940s, he became known as Kid Terror, amending the problem somewhat.
General Glory, DC's parody of Captain America, of course had a sidekick called Ernest E. Ernest, aka Ernie the Battling Boy.
Subverted with the revelation that there have been more than one "Ernie".
Mary Marvel is a borderline case, as Mary is her real first name.
The New Titans (formerly the Teen Titans) had Danny Chase, a character loathed by all fans. Earlier, they had Mal Duncan, who later went through an everchanging series of codenames: Guardian, Hornblower, Herald, and now Vox.
Another Golden Age example, Doctor Hugo Strange [the heroic Nedor Comics version] was known as Doc Strange while adventuring. His costume was basically the 1940s version of Civvie Spandex and he had no secret identity. This was preserved when Alan Moore revived him in his America's Best Comics universe as Tom Strange.
In the first-published issue of the X-Wing Series comics, the pilot Tycho Celchu goes to an Imperial-held planet, puts on a captain's uniform, and reports for duty at pilot barracks in one of the cities, all in order to get intel and be in place to betray them. He does this, however, under his real name and homeplanet. Tycho Celchu, of Alderaan, who defected to the Rebellion after his planet was destroyed, helped keep TIEs off Wedge during the run on the second Death Star, and became part of the core of Rogue Squadron. What's worse is that it worked completely.
The Punisher is often recognized by friend and foe alike as Frank Castle, which he does nothing to dispel...well, except with bullets on occasion.
Doom Patrol has Scott Fischer. Apparently, they gave him the codename Blaze but he never got around to actually using it.
While the Fantastic Four technically have codenames (though not secret identities), they never actually seem to use them. Haven't since the '70s. Mr. Fantastic is universally known as "Reed" or "Dr. Richards". (Johnny Storm isn't actually part of this trope, because his name is actually pretty badass...)
The Runaways tried giving themselves cool codenames when they started out, but everyone pretty much stopped using them as soon as the first volume finished, with the exception of the Cute Bruiser, who insists on calling herself "Princess Powerful."
Another supervillain example comes from Doctor Doom. Everyone knows who he is (what with him being European royalty and all), and considering his status as an Omnidisciplinary Scientist, he more than earns his title. Although since he was expelled from college, so it's more like an honorary degree.
In Captain America #289, Cap's girlfriend at the time, Bernie Rosenthal, spends a backup story daydreaming about being a superhero called "Bernie America", leading to this exchange when she meets her reinterpretation of Steve:
Steve: Uh, B-B-Bernie? I have to t-talk to you... Bernie: Please, Steve—call me Bernie America! You wouldn't want to jeopardize my double-identity, would you? Steve: No, ma'am.
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.
Discussed in an issue of Superman that opens with Superman fighting a guy in a power suit who identifies himself as "Gabriel Van Daniken." Superman tells him that it's the worst name for a super villain and he's ever heard, and Gabriel's reply is "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 a villain to introduce themselves by their real name, instead of calling themselves "Toxic-man" or "Pollutus".
Live Action TV
In the TV series My Secret Identity the lead character doesn't actually have a secret identity.
In practically all versions of The Green Hornet Britt Reed's Japanese/Filipino houseboy Kato goes into action with his boss as Kato.
Alex Rayne of the webcomic Wright As Rayne is public about his identity as a hero, and is a minor celebrity in his home town of Atlanta.
Josiah Brimstone, a mystic hero from the Global Guardians PBEM Universe, fights crime under the codename "Josiah Brimstone". Also, most people think that the telepathic superhero Martini is using a codename, when he's actually just using his real last name (his full name being Benecio Martini). Martini encourages this opinion (by using a tuxedo as a costume and acting very haughty and upper-class), while Josiah Brimstone just doesn't care.
Even though codenames are required at Whateley Academy in the Whateley Universe, Jobe Wilkins just uses... Jobe. Given that his father is the supervillain Gizmatic, aka King Wilkins of Karedonia, Jobe doesn't have a lot to hide.
Launchpad McQuack is often seen with Darkwing Duck as his sidekick ... yet none of the myriad villains, criminals, police officers or autograph hounds seem to care enough to look him up in the Saint Canard phone book. Except one time, when Launchpad was mistakenly reported to be Darkwing himself. Similarly, no one seems to pay attention to Gosalyn or Honker; neither of them uses a code name often.
Justified as one villian referred to them as Darkwing's Fan Club, so people think they just follow him around. Considering that DW's a bit of a glory hound, this is not surprising.
Played with in the new comic series. Negaduck finally figured it out, but only after he saw Launchpad leaving the dry cleaners with Darkwing's costume and Drake Mallard's usual outfit.
On 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.
The eponymous crime fighter of The Dragon Queen doesn't bother with a secret identity. Rather, the Dragon Queen acquired a private investigator license and turns over all criminals to the police for the arrest.
In Young Justice, Artemis Crock uses the superhero name of...Artemis.
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, since Artemis is not a public superhero. The reason, of course, being that they are sisters.
In Super Chicken Henry Cabot Henhouse III drinks his supersauce to become Super Chicken, while Henry's faithfull sidekick Fred doesn't drink anything and becomes SC's faithfull sidekick Fred. But then, he knew the job was dangerous when he took it.
Played with in Super President, where his super name is his real job, but he still manages to preserve a secret ID, somehow...