A fairy tale is a story that is passed from person to person by word of mouth, a story that consistently changes when it is retold. Whereas normal literature is recorded physically, and is essentially immutable, by the time any given person hears a fairy tale, it likely bears signs of any number of people's additions. Though fairy tales may have their genesis in one specific part of the world, they resonate with people in many profound ways, especially since they are intended for children, and so are read to children. They may act as ways to teach lessons to a child, or a child may find parallels to problems in their life in the story, but either way, when the child is grown, it is likely that some fairy stories will be at their core. So when these people move around, they take the stories with them.
This essentially becomes a requirement for fairy tales, then: they must be a form of literature comprehensible to children, and in fact one that they can seek greater meaning from, whether it be in the form of Aesop-like morals or in that of particularly meaningful symbols. This allows the children to build a love of the stories, and in turn pass them throughout the world. A fairy tale not designed to be accessible to children is then, at its heart, not a fairy tale. A fairy tale that recounts real events, blurred by time, is also not a fairy tale, but a folk legend. Though fairy tales may have their start in these legends sometimes, they can only be considered to have ascended to the status of fairy tale when they spread beyond the region where they are born, and when they become an abstraction of themselves, where real emotion and probable character action fly out the window in favor of action and fantasy.
It is also, as I mentioned before, deeply important for a fairy tale that it not belong to any one storyteller or place of reference. Fairy tales must be relatable, and often, those who retell them will alter them a little bit for the listener or from their own experiences. This allowance for retelling and changing is actually crucial to fairy tales, for I would say that defining a fairy tale as a story which is transmittable by word of mouth and utterly protean would be the simplest possible definition. This is why, if you ever discuss a story, say Little Red Riding Hood, with someone else, you may find yourself having a disagreement as to how the story goes. Everyone assumes everyone else knows these basic stories, but one person might have heard the version where everyone was eaten by the wolf, and another where they were somehow saved by the woodcutter, or another where Red simply escaped. It is this idea that every telling is slightly different that is at the crux of these fairy stories.