Bucky, Captain America's Golden Agesidekick and later 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 whether 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".
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.
Doctor Doom. Everyone knows who he is (what with him being European royalty and all), and considering his status as an Omnidisciplinary Scientist, he lives up to his self-appointed title as "doctor" even if he was expelled from college.
While the Fantastic Four have code names, they've never had secret identities, and are frequently called by name. In particular, Mr. Fantastic is universally known as "Reed" or "Dr. Richards".
Luke Cage hasn't been called "Power Man" in years. But then, he doesn't have a secret identity, either— even calling himself Power Man was just for publicity purposes. Granted, "Luke Cage" isn't his birth name. He got his name changed after breaking out of prison.
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.
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."
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. ("Marvel Woman" was tried out for a while, as did continuing to call her "Phoenix" even when the actual Cosmic Entity wasn't around. Neither stuck.)
Jean eventually started using the Marvel Girl name again in X-Men (2019).
Her Kid from the Future Rachel is also usually just Rachel (though she also went by Marvel Girl for a while, and also Phoenix and, for some reason, Prestige.)
Similarly, it turns out Zaladane's name is Zala Dane. (We think. At one point. Maybe.) 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)
And the name Polaris was first given to her by a mind-controlling villain after a long run being just "Lorna Dane," then retiring, then being dragged back in by this incident (although Classic X-Men incorrectly had her using it before this.)
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".
Discussed in the tie-in comic. "How come everyone has a codename but Rogue is just Rogue?" "Same way Jean is just Jean." "But why is-" [alarm sounds, everyone runs off, subject is never brought up again]
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. Also, considering her past non-supervillain criminal activity, it's not like the name "Emma Frost" is associated with an innocent civilian Secret Identity she wants to protect. It's probably better to have people know that Emma Frost is both on the side of the angels now and not to bemessed with.
Like Jean, Emma also started using her name again in X-Men (2019). She's actually back in the Hellfire Club again, but this time the organisation is allied to the X-Men.
The whole Hellfire Club falls under this, especially because the members change, and many have operated independently before/after their time in the Club. It's just easier on the characters and the reader to say "Oh, Crap!, it's Sebastian Shaw!" instead of "Oh, Crap!, it's the Black King! Uh... the first one. Wait, who's Ned Buckman? I mean the other first one... Quentin who? What do you mean there's a London branch?..." and so on.
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.
X-Women tend to lose or not have code names (or have code names that are for all intents and purposes their real name, like Rogue and X-23), possibly because their creators really like the real names (Pryde, Frost, Grey) or introduced 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." Jubilee is a borderline case - it's short for her real name, Jubilation Lee, but it sounds like a codename).
