You can actually save screenshots in this game, I can confirm.
So the problem could be that maybe you are using external program like Rivatuner, Nvidia GeForce Experience, AMD Catalyst or similar, some of which may conflict with the GOG overlay.
Other options could be: for some reason the overlay does not start when you start the game. You remapped the F12 key to another key, or maybe the path to save screenshots has some problem (the folder in read-only mode, for example).
Hope you get lucky with the problem.