You save using the diary on the bedroom.
Or the energy balls on the dungeons.
Viewing post in Mahō Shōnen: Magical Battler comments
There is no way to prevent the save from breaking if you are using an incognito tab.
The game should be fully playable on a phone's web browser if that is of any help (if you decide to do this, make sure to read on the game description how that works, because navigating out of some menus is not intuitive).