Fictional characters necessarily have a very difference perception of self hood than real people. Unlike them, characters are less distinct entities and more so concepts given life. They may be very specific concepts, but they are still concepts and ideas. Even the simplest character can experience the sensation of not just one self but many selves, all a part of them and to be mediated.
Adaptations are one obvious source of complication since they can greatly differ from the original work while still portraying the same characters. At the same time, they can also be incredibly similar, being perhaps too faithful to the source material. This variety means that there's almost as many experiences of adaptations as there are works adapted. Generally, an adaptation is either subsumed into the source material or leads to another version of the same character being created.
Whether an adaptation creates a new incarnation or not, the source material is still linked to it. They can access it as though it were their own memories, because in a sense, they are. When an adaptation is subsumed into a work, that's all this is; a new set of memories that are slightly different versions of what happened before. If the adaptation is significantly different it can become its own being in a process referred to as splitting. The multiple independent versions of one character are each referred to as incarnations. The source materials of incarnations tend to use familial language to affectionatly refer to each other: an adaptation is their "kid," "son," or "daughter," while the source material is their "parent."
All incarnations of a character share a psychic connection, of sorts. They can't communicate telepathically, but they do have at least partial access to memories of the other's stories and just generally understands them. This connection can be hampered by poor self awareness, but even the most impotent characters can still get glimmers here and there. Especially in cases of serial media, this can lead to the disorienting feeilng of having ghost memories, where characters can recall moments from your past that you did not experience and do not yet have an equivalent in their adaptation.
Usually, incarnations are all contained or classified under the original version of a story. All adaptations of Moby-Dick, however lose, would eventually flow back to those characters as written by Herman Melville. But in some cases the way a character is portrayed by popular culture becomes different enough from the source material for the two to become different characters. In these cases the first recording incarnation is referred to as the "original" or "ur-" version, and the incarnation they're all subsumed under is the "archetypical" version. One incarnation influencing another indirectly by changing their perception in the broader imagination is referred to as bleed.
Translation is technically a form of adaptation, but it very rarely results in new incarnations splitting off. It does increase the number of languages a character can speak. Characters can only speak languages they are told to know in stories if they are given a translation in that langue; they can speak in any tongue that their story has been given form in fluently. Usually this covers a direct complete translation, but even a detailed summary could do the trick. Characters often experience a sense of dissonance between their selfhood when speaking different languages similar to the dissonance between the original and an adaptation. It's still them, undoubtedly, but there's something different about it. This unease, and hysteria about mistranslations, is enough to keep several characters as linguo-purists that only speak the language they were originally created in.
When an alternate universe is included in the plot of a story it results in a "lesser" incarnations, no matter how similar the two alternate universes are when portrayed in story. This is distinct from adaptations because the story as written has alternate universes as a part of it. While two dictinct versions of the same character are created, these incarnations are considered lesser because they usually lack the psychic link that defines incarnations created by pop culture or adaptation splitting. Those cases are sometimes referred to retronymically as "greater" incarnations because of this. Some cases do count as greater incarnations, such as the existential situation of most superheros. They have an archetypical incarnation which shares a psychic link with alternate universe versions of themselves created by the story. However, it's easy to make the case that this is caused by adaptations and pop culture and not the existence of alternate universes within the stories as written. Therefore, the situation is generalized as
Greater Incarnations = splitting caused by pop culture and adaptation
Lesser Incarnations = alternate universes written as such in story
Fanfiction does often involve the creation of alternate universes and can serve as adaptations, but it almost never results in splitting. Fanfiction is nigh-universally unofficial and explores what-ifs that aren't expected to have any bearings on the canon plot. They are exploring subsections of existing characters and are generally based on pre-existing attachment to an existing character and are percieved as an extension that character instead of a retelling of that character. Thus, they are almost always subsumed under existing characters instead of creating incarnations.
That isn't to say that fanfiction and fandom can't have an affect on characters or create incarnations; they can. It just takes the same amount of effort to change perception through fanfiction as it does to change it through other means and fanfiction has far fewer resources. An AU that becomes very popular or drifts enough from canon portrayal can create an incarnation if it gets popular enough. Because of the effort this takes, it's a pretty rare phenomena. Usually fanfiction is just something that a character experiences through dreams and has a new part of themself that they can remember from.