I was wondering how viable would this be...
When asked about backing up your game you could have 2 options: backup, don't backup.
On backing up a copy would be made and added to a folder named Copy game name...
This would be better than manually dong it.
What do you think?
That's pointless. If it is just a different folder on the save drive, with a disk crash or stolen laptop you will loose both at once.
And for keeping track of your changes you should use source control (e.g. git) anyway and either have your repository on a server or backup the whole repository to an external drive.
IMHO this may be done with plugin, there's already a source control integration API for plugins, that may let it know which files to backup.
What I usually do when I'm in extra-careful mode is just create a batch script that compresses the entire working folder and copies it to a backup location, like an usb drive or cloud storage.
You could add the shortcut to the batch file on a taskbar or start menu, and it becomes really streamlined that way.
Save your work, then click on taskbar icon, and it's done.
That's the hoarder method. Not really useful for source code versioning, but quite effective for safety storage, and art/binary backup.
The smarter method is to use git / github, or other version control. But that's boring. :P
The lazy method is to have your game folder in a Dropbox folder, automatically synced.
And the paranoid method is to combine all of the above.
I have a local git in a folder and the folder is synced to Google Drive using Insync (it's really similar to Dropbox, but a lot cheaper). This actually works surprisingly good.