Windows or Mac?
in my view this is not a DW-thing. I put an icon on my desktop (I use WIN, so I only can speak for it) and so I'm able to use/change a DW icon by right-clicking on the icon (screenshot from my German WIN):
I choose "Anderes Symbol" ( > another symbol) and I reach this menu item:
where I can definitely select the appropriate icon.
Well, ive realised its just in the start menu, list, viewing the folder directly and the icon shows up, what is wrong with this PC!?
If you right click the shortcut and click "Change Icon", you can select the proper icon and click "OK" then "Apply". It will have the green Dw icon then.
I can only guess as to what's wrong with the PC.