If you login with your account through the app, it should show all purchased games there, it would just ask you to re-download them (but it would remember you’ve bought them). An exception would be the big bundle that Itch had a few months back, I’m not aware of the details, but it had some special handling.
If you login with the same account that bought a game, and it doesn’t show as bought in the app, it sounds like a bug to me.