So there are games that are different in behaviour, when downloaded from the website compared to being installed by the itch app?
This ... does not sound quite right. There is a mission statement somehwere, that the itch app is considered optional, and this would make that statement void. Is your app communicating with the itch app, when running, or how does this work? Or is this about html5 games? What am I missing?
I am just asking this dumbly, because I never encountered such a game here. Those type of user authentication are typically not needed for most indie games. If it is for some online services, ok, but otherwise it is just drm. Most games here are offline ready to go unzip and play drm free games.