In Europe, fairytales have various different names of the same story, or variants. One example of this is Charles Perrault's Cinderella ("Cendrillon ou la petite pantoufle de verre") and the much darker, violent version of Cinderella by the Brother Grimms ("Aschenputtel"), where the latter story involves mutilation of the feet and the eyes of the step-sisters and the step-mother being plucked out.
However, Charles Perrault has another similar story known as Donkeyskin. The plot is a bit different, but the main premise of an object that only fits onto the specific person (destined to marry) still remained. A similar variant exists as "Catskin" in England by Joseph Jacobs.
The story of Cinderella isn't just limited to Europe. One of the earliest version of Cinderella actually comes from China, under the title of Ye Xian. There's a version in Korea known as Kongjwi and Patjwi, and variants found throughout Africa.
Although Cinderella is probably one of the most famous examples of this, it isn't the only one: See Little Red Riding Hood, The Honest Woodcutter, any many others.
How can so many children's stories be so widespread?