The Cinderella story is a quasi-universal folk-tale with innumerable variants across cultures (it's tale 510A in the Aarne–Thompson–Uther Index): in its most usual variant, it combines a mistreated stepdaughter, beautiful clothing given by a supernatural being, a ball where a prince fall in love with her, and an identity test based on a shoe. The recent version of Charles Perrault (1697) is the best known in Western cultures and includes the glass slipper.
In 1893, folkorist Marian Rolf Cox identified 345 variants of the story, and noted that the glass shoe/slipper variant could be found in only 6 of them, who "have evidently been subjected to a French influence, and that at a comparatively recent date" (p. 506). Otherwise, Cinderella shoes are made of different materials, which is often gold, but often left unspecified, and in some variants the "proof of identity" is another object (a ring for instance).
The glass/fur debate is actually a French one, due to the homophony between verre (glass) and vair (an obsolete type of fur). The earliest recorded mention of the debate is a review by Théophile Gautier of 1839 about a production of Rossini's Cenerentola, where he said (cited by Hoffmann, 2016):
The slipper of vair (and not glass, a matter hardly suited to slippers), seems to us to be a dramatic device not inferior to that of Othello’s handkerchief.
This opinion was reiterated by Honoré de Balzac in his novel Sur Catherine de Médicis (1841): in a paragraph where a character describes how fur was a luxury item in 15-16th France, Balzac writes that Cendrillon's slipper was "without doubt made of menu vair" and not of verre.
Since then, there has been an ongoing debate in France (and later in other countries) about whether Perrault made a mistake (eg he was told vair when he collected the tale and wrote verre because he did not know what vair was) or did it on purpose. Some later editions or versions of Perrault's tale substituted vair for verre, and this issue has found its way in the public consciousness as some fun "Did you know that..." anecdote.
However, Perrault's tale is unambiguously titled Cendrillon ou la petite pentoufle de verre. Perrault was Louis XIV's go-to guy for arts and letters (and a member of the Académie Française), so he knew a thing or two about French language. There's little doubt that he meant verre or that this not was a last-minute correction by a printer concerned with proper spelling (the book was published when he was alive): the stepsisters cannot put the shoe on at all, despite "doing everything possible", which makes more sense with a glass shoe than with a fur-made one. Also, a fur slipper does not make much sense either: in Furetière's dictionary from the late 1600s, a pantoufle is, like today, a comfortable shoe made to be worn indoors, not something worn at a ball. And a fur shoe?
So basically we do not know why Perrault used a "glass slipper" rather than a "golden shoe", and we can only speculate that he did this to add a layer of fantasy and enchantment to his tale. Hoffmann (2016) writes convincingly about the perception of glass and crystal in Perrault's time and how these materials were used both in reality (luxury objects, often strange ones) and in fantasy tales (caskets, palaces and grottos), both for their beauty and symbolic value:
The notion of a glass shoe was hardly shocking or even particularly novel; drinking glasses in the form of gilded riding boots, or the footwear of a Roman soldier, had been part of the repertoire of Venetian glassmakers since the late sixteenth century. [...] Charles Perrault was not terribly voluble on fairy details, so I will need to leave it to the reader to imagine what Cinderella’s glass slippers might have looked like in the mind’s eye of Perrault’s seventeenth-century readers: etched rock crystal or the sparkling new glass that vied with it; a Venetian affair decorated with millefiori; or perhaps like Louis XIV’s table, bearing nude gods and fanciful hunting scenes. For Cendrillon, Bernard Perrot’s ruby red glass made with deadly arsenic and precious gold would work particularly well. It would remind the prince and the princess of the dangers that may be lurking in enchanted glass things — tunnels or shoes — even when they bring fairy-tale lovers together.
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