Yes it'll copy the screen...but then you just use an image app like MSPaint or MS Photo Viewer or whatever and crop out everything but that icon and save it as a jpg or GIF.
You can also get a larger version by doing similar to what you do on the Mac. Go to the properties of the shortcut, and click the Find Target button. Then go to the properties of the executable and you get a larger version of the icon. Print screen that and crop it out if you need a higher resolution version of it.
There is no way in Windows to change the icon for an actual executable, because the icon is usually stored in the executable itself. You can change the icon for a shortcut however, in the Properties screen is a change icon button. If you want to use a standard icon, the shell32.dll file that usually comes up has many, or the icon from another executable can be used by browsing for it and selecting that icon. However, note that if you ever delete that executable, the icon is deleted too as it is stored IN the executable.
You can also download an icon file creator. These can create different types of icon files which Windows can use, so you could use the printscreen cropped icon to make a new icon file, then select that as the icon for whatever you need.
Of course if you are just looking to get a copy of the icon for use on a webpage or something, then you can just make the image file.