Why is there a koozie on this bike's chainstay? A friend of mine recently purchased a used hardtail mountain bike. As he showed it to me, I noticed that there's what looks like a koozie wrapped around the chainstay, towards the rear axle.

The branding on the koozie is the same as the make of the frame (Stevens), and it's being held on by velcro. It doesn't appear on the manufacturer's website as far as I can tell.

The outer layer seems like it's nylon, and there's some minimal padding.

What's the purpose of this thing, and what do you even call it?

I've included a photo for reference.
That is a "chain stay protector" and it is installed in, let us say, a non-traditional way.  Typically it would be a couple inches forward of where it is.

As the name suggests, the purpose of the pad is to protect the chain stay from wear and damage from the chain, which can hit the stay with a lot of force when riding at high speed over rough terrain.