I didn't quite manage to fix the issue, but at least to overcome it. Let me explain: originally, I downloaded the Recalbox OS v4.1.0 from https://github.com/recalbox/recalbox-os/releases/tag/4.1.0 This version has the issue with not being able to save the settings. This version is installed by simply copying a bunch of files to the SD card and letting the RPi boot.
But then, I discovered the version 17.12.02 of Recalbox, that I downloaded from https://archive.recalbox.com This is an image (img.xz) file, that I installed using the Etcher software (as mentioned on the download page) on the SD card. This version of Recalbox seems to do the trick and I was able to enter several settings that were still set after rebooting the RPi.
I hope, this helps everyone with the same problem.