This was added a few years ago...
In the Classic Client, go into Options menu on your paperdoll
Click on the "Interface" Button (Right side, top- looks like a computer mouse) in that pop-up
Scroll ALL the way down to the bottom in that menu
You will see
"Select Character Backpack Style"
The choices are
Default
Suede
Polar Bear
Ghoul Skin
Typically only your main pack, plus one other, will use the non-default skin you choose, with the rest being default.