Unfortunately, so long as you play the web version, that's entirely down to the browser and your device. I recommend either manually exporting/downloading your save using the options provided, or playing the desktop version of the game. Also, if you'd like a head start on a new game, there's a save-bank channel with a variety of saves at different points in the game.
I believe the itch.io app is rather parsimonious with it's cookies, so most of the web versions of games you play through the app will not do a good job of saving data in cookies... can't confirm but I don't have any problems using it to install web games and then losing data so I'd recommend doing that! (They're usually pretty small after all!)