These superheroes just aren't called superheroes. They often don't wear costumes or use code names, but they have abilities far beyond those of normal men, and are superheroes in all but name.
Occasionally, such stories will lampshade the trope by having characters in off hand discussions about whether they'd look good in a cape, or using Something Person-style nicknames, but discarding the ideas as being "silly." A "This Is Reality" remark can be thrown in, as well.
Like many tropes this one has underlying practical considerations, such as:
Aesthetics: Most classic comic book-style superhero costumes tend to look very silly in live action, which is why Movie Superheroes Wear Black. Plus, with a few relatively rare exceptions (such as Christopher Reeve as Superman, Lynda Carter as Wonder Woman, or Scarlett Johansson as Black Widow) it is very difficult to cast someone who physically resembles comic book interpretations (especially in cases where body shapes are exaggerated).
Genre Shift: When the creator is actually aiming for the story to look and feel as if it belongs in Science Fiction or Urban Fantasy categories, and does not initially realize that the story fits the conventions of the super hero genre. They may or may not act kindly to people pointing out the similarities.
Marketing: Some creators may wish to avoid their characters being seen as superheroes in order to prevent incorrect expectations of the work. The marketers may also want to play up the popularity of their lead actor or actress, meaning that a face-covering mask is a big no-no.
Budget: Mostly affects TV shows rather than films. Most live-action shows have to work within very tight budgets, so it can be difficult to create a faithful translation of a comic outfit without it looking cheesey or low rent. The Justice League of America pilot and the later seasons of Smallville are rather infamous for having comic-accurate but extremely cheesey looking costumes, for instance.
Legal: Licensing for intellectual properties can be very complicated and they may be unable to use the iconic costume of the hero but can still use the character itself. A well-known example is Smallville where they had the rights to use Clark Kent and some back story elements but not have the rights to show Superman (for most of the series).
Compare with Sci-Fi Ghetto, Not Using the "Z" Word, Animation Age Ghetto. See also Civvie Spandex and Spandex, Latex, or Leather. Series in which people are Not Wearing Tights typically use a Differently Powered Individual label.
