I have this strange bug in which the gog overlay freezes a few seconds after the game starts. The "Now playing" notification is left there stuck, or, if I manage to open the overlay before that, the whole overlay gets stuck in the screen. The only way to play without any freezed GOG stuff (outside from disabling the overlay) is to rapidly open and close the overlay as soon as the game starts.

Interestingly, I faced a similar problem with Day of the Tentacle Remastered.

Found a partial solution. If you change the game to windowed and then back to full screen, the notification does disappear correctly.