Yes, the desktop.ini inside s folder controls the icon of a folder. The path to the icon can be relative (such as myicon.ico) or absolute (such as C:\customicons\icon1.ico). If it is absolute, it is likely that on your friend's computer the file with the icon is not present.
The solution will be to copy the icon to each folder and use relative path in every desktop.ini. Or your friend must install the icons in the same location as you.