That key can be used in various incarnations of your app's UI, on the web, across platforms, or even in system services (like notifications or Siri's sheets) to let the user quickly make the association with your app.
leather does not add distinctiveness to your app, it creates an association between your app and leather that now shares a bond with the 100s of other apps that use leather. the only apps i can think of that have used leather to such an abstract, memorable extreme are paypal (blue leather wtf?) and find my friends, the classic.
mental models have nothing to do with this, unless the average user has a previous experience that associates your product/service/offering with leather. even cowboy boots, even some sort of evocative mood, whatever, that's fine, but if you are using leather because you think you're making it distinct, there are far too many others doing the same.
Also, I didn't mean to imply that leather in itself teaches you how an app works. However, once you've used an app with a distinct skin, the next time you launch that app the skin, as a sort of visual touchstone, will help you remember how you used it in the past, etc.