Hey, I am now in Windows, I copied my save folder from the game folder in Linux into an USB stick, installed the game in Windows from the Itch app, then opened the game folder which was empty except for the Itch folder.
Then I launched the game then opened the game folder again.
Now it was full. I don't know why but the game apparently needs to be launched at least once in order to create the folder, this is an useful info to know which might solve some empty folder issue on a fresh install. ;-)
Then next, I copied my save folder from my USB stick into the game folder in Windows and launched the game again.
It works! My saves from Linux are now present in my game on Windows and I am able to continue in Windows from where I left in Linux! :-D
This means that the saves are not system specific, they are universal, we can continue a saved game from and on any system, this is also an useful info to know so I share it with you. ;-)