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I've read somewhere that there is this writing rule stating that, for some superpower, it would be less believable if two completely different settings are present in order to obtain the same superpower. In the article, I remember it mentioned in the movie Spiderman (which I've not watched), the protagonist and another character gain the power through different means.

What's this rule and does it have a name?

First, welcome to StackExchange!

Now onto your question: there aren't any enforced rules when it comes to superpowers or even fiction. The closest thing would be a consistency guideline. Consistency, while not a rule, is usually something a reader will be quick to call out if they perceive it to be broken.

When you hear readers complain about a story having many plot-holes, a lack of consistency is sometimes the cause. But what the reader expects to be a constant changes from story to story. As the writer, you set what is believable or unbelievable in your setting.

Take My Hero Academia (Boku no Hīrō Akademia) for example: the story establishes early on that people in that universe are born with superpowers. Their genetics determine what 'quirk' they are born with and from there they can train their power and become stronger, but never change the ability itself. Without getting into any spoilers, the plot deals with a rare example of where that isn't the case. While that may break the consistency established at the beginning of the story, the plot goes on to explain how this happens and what it means in that universe.

The bottom line is: As an author, you determine everything that is a possibility in your world. As long as you give the reader enough information as to how anomalies can occur (or even foreshadow that they may occur), the consistency isn't broken.

I have been researching comics history for a few decades and I have never heard of such a rule. Others in the thread have given examples.

It's true that there are a few stories in which all superpowers have a common source, typically an alien contaminant into the Earth biosphere (J. Michael Straczynski's series Rising Stars and Supreme Power both explore this option, as does the Wild Cards anthology), however these are the exception rather than the rule.

Keep in mind that the most established superhero universes (Marvel and DC), the characters were created in tandem and only after the fact organised into a unified 'universe' ... so you might have the Human Torch (who flies because, I suppose, heat rises) battle the Sub-Mariner (who flies because he has little wings on his feet) ... and there was never any sense of contradiction there.

For something to genuinely be considered a "rule" of writing, which will delineate bad writing from the rest, it generally has to denote something that is hard to do successfully. Given the fact that most Superhero universes violate this rule all the time, and they're currently quite popular, evidence for this rule being able to separate good writing from bad is pretty minimal.

So odds are good this was just a statement from someone who prefers that sort of thing, aggrandizing their own preferences by declaring them to be a "rule".

That having been said, "believe-ability" is an interesting domain. The thing is, in the general Superhero milieu, pretty much anything which does not directly contradict previously established rules is more or less acceptable. To the extent that Superheros are science fiction, they are very clearly on the softer side of it. The softer the sci-fi, the less "believe-able" it needs to be; the audience is pretty willing to swallow a lot of oddball concepts if they are interesting and you're giving them what they're interested in (cool action/etc).

Sci-fi hardness is a useful way to examine this "rule". It is effectively promotes a kind of "One Big Lie (TV Tropes)" style of superpowers and world-building. That is, you pick one, or maybe a few, mechanisms for how superpowers work, and you stick with just that. Different powersets are just a matter of how someone chooses to use the special rules and physical laws you create.

Is that more "believe-able"? Well, it is easier to swallow a single change to reality than the anything-goes style of many superhero universes. Some people may take a work more seriously if it is like reality except for one thing, particularly those who like hard Sci-Fi. And having this kind of focus can lead to some interesting worldbuilding scenarios, which can lead to unique superheroics and the like.