That's not so easy to do because $file['file_name'] stores "name.jpg" and you wouldn't want "name.jpg_(domain.com)". Besides, if someone downloads 10 images and then views them in a file manager, like windows explorer, they will all be grouped together since they start with the same prefix. This would make it easy to spot them amongst 100 other files in the same directory.
My advice would be to find all the places in download.php where $file_name is used and make adjustments as needed. This is hard because if the file is a remote file, retrieved via remote URL, you have to manually parse the name. It's also possible that some of those files might not have a file name extension. Good luck.