I’m gonna be honest, I tried unlimiting browser cookies and I still lost save data, it seems to be more a problem of how itch handles it than just how the cookies are stored.
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!)