"Is he strong?
He's got radioactive blood
Can he swing from a thread?
Take a look overhead
There goes the Spider-Man
- Spider-Man Theme Song
Great question, and one that shows that philosophical analysis can be applied to just about anything, since philosophy is a method, and not just a list of topics! To answer your question, in the broadest sense, you're dealing with categorization. From WP:
Categorization is the ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such as objects, events, or ideas), organizing and classifying experience by associating them to a more abstract group (that is, a category, class, or type),1 on the basis of their traits, features, similarities or other criteria that are universal to the group.
In the example of spider-people (and don't forget spider-pigs!), there are indeed conditions needed to be in the set of spider-people. Having been bit by a radio-active spider, being from Brooklyn, being a teenager, having a love interest, and slinging webs all seem to be the sort of conditions that allow for a fictional character to be a spider-person in their respective universe. Some of these conditions are stronger than others. Such requirements are often referred to as necessary and sufficient conditions. From WP:
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P. (Equivalently, it is impossible to have P without Q, or the falsity of Q ensures the falsity of P.) Similarly, P is sufficient for Q, because P being true always implies that Q is true, but P not being true does not always imply that Q is not true.
In an alternative way to categorize, derived from a philosophical idea called family resemblance, instead of necessary and sufficient conditions, membership in a category is based on resembling strong examples where no condition is absolutely necessary. From WP's article on prototype theory:
Prototype theory is a theory of categorization in cognitive science, particularly in psychology and cognitive linguistics, in which there is a graded degree of belonging to a conceptual category, and some members are more central than others.
Thus, in the case of a spider-pig, the reason that a spider-pig can be a spider-person, is that while pigs are poor examples of people, all of the other conditions are met, such as English speaking, having an Uncle Ben, wearing a blue-and-red outfit, etc. Certain words like 'game' exemplify how sometimes some members of a category may not share any properties, but still be part of the same set.
As for the multiverse, a popular science fiction trope, that invokes the notion of possible world semantics. From WP:
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their metaphysical status has been a subject of controversy in philosophy, with modal realists such as David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not.
That is to say, we can talk about 'worlds' as being similar in some regards, and then speculate on individuals, facts, truths, and other aspects of these worlds. As noted, some philosophers argue that possible worlds are actual physical worlds such as modal realism describes, while others believe that such words are just that words which describe ideas and not reality itself as nominalists claim.
In physics, there is non-empirical conjecture that constitutes metaphysical speculation called the many-worlds interpretation. From WP:
The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe.
So, the spider-man multiverse or any What-If-like story lines revolve some basic philosophical ideas that are generally covered under the labels of linguistic modality and epistemic modality.