Maybe you can provide a backup feature, just in case they use another computer, or clear the browser data by accident. For example, open a text box and show the saved string and allow the user to copy them. Or download as a file that can be imported later.
Also localStorage is like cookie, so if you store something on a.itch.io/game1, you can't acces it from b.itch.io, but you can access it from a.itch.io/game2. It depends on the domain name of your linked game.