Yes, if you log out in the HamAlert app and then log in again, the notification sound gets reset. The app wasn’t designed with frequent logging out in mind. Is there a reason why you need to log out and back in? I keep my app logged in all the time and have done so for years.
If you simply mean restarting the app (without explicitly logging out in the menu), then the notification sound should not be reset by that. At least on my iPhone, the Morse setting persists even if I kill and restart the app, as it should.
You can have multiple phones logged in to the same HamAlert account, and all of them will receive the same alerts, so there’s no need to log out. If you want to temporarily disable push notifications on a specific device, there’s a toggle switch for that in the app’s settings.