> But please write more how the icon is set.
Like I said, I am not the original developer, so it is hard for me to give more details about how the icon is set. However, I was able to solve the issue myself and find two things.
For one, it is not the fact that I put the EXE in the "Program Files" folder that made Windows show the feather icon; it is because I also added a shortcut to the application in the Windows 10 Start Menu, causing Windows to retrieve the feather icon. Removing the application from the Start Menu would restore the correct icon.
For two, I also found out using Resource Hacker, that the embedded icon in the EXE had various sizes (16x16, 32x32 and 48x48) but that there was also a large feather icon of 256x256. I deleted the large feather icon and it completely solved my issue. It looks like Windows, for the Start Menu, favoured the 256x256 icon and therefore picked this from the EXE.