Earlier this week, I was studying Saul Kripke's Identity and Necessity, in which he sets out an argument as to precisely which relations of identity are metaphysically necessary. Briefly, he divides all names and descriptions into "rigid" and "non-rigid" designators: rigid designators, such as proper names and technical terms such as 'heat' refer to the same thing across different possible worlds; non-rigid designators, on the other hand, refer to different things across different possible worlds. (This is not to be confused with different worlds using different words to refer to the same thing - in Kripke's system, "Andrew" refers to me even in worlds where I am known as "Christopher" or "Timothy" or even "Jessica").
According to Kripke, any identity statement in which both objects are referred to by rigid designators is necessarily either true or false. So "Andrew is the son of Tim" is necessarily true for all worlds where I exist, and "Andrew is the brother of Chris" is necessarily true for all worlds where both my brother and I exist; however, "Tim is the editor of Overdrive" and "Andrew was the Chess captain 2010-12 at Camp Hill Boys" are merely contingently true - that is to say, there are possible worlds where they are not true.
As it happens, I do not agree with Kripke's view of personal identity - he relies on the assumption that there is something which is irreducibly "me", whereas I would tend towards viewing objects merely as the sum of their properties. However, suppose he is correct regarding identity of real-life people. Does this still hold for fictional characters?
One old trope of fanfiction is to change who one or more parents of a character is. This is mainly done for shipping purposes - for example, making Harry Potter the son of Snape in order to ship Snape/Lily, or removing the sibling relationship between Elsa and Anna so that "it isn't incest". Except that, under Kripke's theory of identity, it is incest and there's nothing you can do about it without making them two fundamentally different characters.
Under my preferred theory of identity, this is not the case since the characters in a fanfic cease to be the same characters as those of the original as soon as even the slightest difference appears between them. I'm still a bit concerned as to precisely why one would want to ship Elsanna, but I can sort-of allow that "it isn't incest". (Even if it was, that would be far from the worst relationship going on in fanfiction - that 'honour', within the bounds of what I've encountered, would be Harry/Fem!Harry fics...)