Greetings. Itch does not have any API or cloud system for saving games.
Each game and developer is responsible for the save code of their games, as you will understand, each game is different. If you have a problem with a game, it is best to consult directly with the developer of that game.
Web games usually use 2 ways to save games, either they use cookies or they use localstore. In both cases, the game is saved on your computer and not in the cloud.
You mention that this happens to you with all games, that makes me think more of a configuration problem on your end. Maybe some security or privacy setting that clears your localstore.