While fur being a variable material* in clothing would be incredible, that's outside the scope of this post. Instead, I would speak of them being a variable ingredient as found with different ingredients and the output changes in cooking.
It was the fur cloak and the fox hat that made this idea occur to me, and it started with the idea that it would be nice to have a "Fox Cloak" or "Badger Cloak" instead of a "Fur Cloak", perhaps with higher % of survival gilding being available due to the (marginally) more difficult task of getting 10 pieces of fox fur instead of a myriad collection of whatever.
In short:
I would like to see animal hides have a direct impact on various elements of the final product, including leather.
Making armor out of rabbit leather? Better expect it not to be as durable as that made from mammoth leather. Perhaps Badgers and Fox leather provide different levels of deflection vs absorption.
That Fur Hat you're making? It'll give different base stats (ala the Fox Hat) dependent on what you make it out of, and perhaps even require different gild stats (and have different gild capabilities) than if you chose different materials. Extend this logic to hemp, silk, linen, etc and see what unfolds as clothes and gilding become even more involved, even more diverse than ever before!
In short: Make tailoring even more engaging by making every material used alter the end product. (Yes, including the choice to use cattail fibers, flax fibers, hemp fibers, stringy taproot, stinging nettle, etc for string).