That's really strange. I always use chrome and saving has always worked. I was thinking that it could have been because I pushed a new update last night, but I just did some testing on an unlisted itch.io page and my progress transferred between the old version and the new version. Maybe you somehow cleared your browser cache?
Can you verify that saving has ever worked for you? Play a couple levels and then close the window and come back and see if your progress is there.
Hmm, I don't know what the issue could be then. If you remember what puzzle you had gotten to, I can send you a save file with the unlocks up to that point so you can get back to where you were (it would have to be for the downloadable version though, idk how to manipulate the save file in the web browser)