First, I have to say that I'm not playing on Windows but on Linux. Syndicate being a DOS game, though, I don't think my problem (and the fix) is related to the OS used.

So instead of installing the game through the GOG installer, i extracted the files (with innoextract) and ran the game manually through DOSBox using the config files made by GOG team.

The game works fine except that I can't save: when going in the load/save menu, choosing a slot, changing the name and pressing save, nothing special happens (no error) and when going back in that menu, the savegame is not there anymore.

I fixed the problem by creating a folder named "SAVE" in the game dir. Looks like Syndicate is unable to create it by itself for some reason. And I don't think it's specific to the way I manually run the game as I've seen another thread where someone on Windows has the same problem.

Maybe GOG should update the installer so that the game folder includes a SAVE subfolder? It would fix the problem once and for all for the players that encounter it.

I know it's an old thread, I replied to thank you for your suggestion.
I was loading the wrong .exe, MAIN.EXE instead of SYND.EXE. Sorry for "necroing"! I hope this is useful for future reference because it looks like saving is a very common problem with Syndicate Plus.