Jewish law does not require a fur hat, does not mention one, and could not, since the garment is a product of eastern Europe rather than of the sources. The shtreimel is custom in the strict sense, and it is custom whose beginnings are not documented.
Several accounts circulate. One holds that Jews adopted an item of Polish or Russian noble dress and kept it after the nobility had moved on, which is a common pattern in the history of dress. Another holds that a decree once required Jews to wear an animal tail as a mark of contempt and that the community turned the humiliation into an ornament of the Sabbath; no such decree has been produced. A third simply reads it as the honouring of Shabbat with the finest thing available in a cold country. None of these has documentary support of the kind that would settle the matter, and the honest position is that nobody knows.
What can be documented is later and more interesting: the hat is a marker of community identity in a way that no source requires and no authority imposed, sustained entirely by the communities that wear it. The recent arguments over synthetic fur, driven partly by cost, partly by animal welfare legislation in some countries and partly by the difficulty of obtaining pelts, are being conducted on exactly that footing, as a question of custom rather than of law.