Deliveries includes an option to update in the background, even after you quit the app. This works by running a separate app in the background, named “Deliveries Express”. (In some earlier versions of Deliveries, the background app has the same “Deliveries” name as the main app.) Even if you have background updates turned off, Deliveries Express handles certain features like presenting notifications. Deliveries Express also manages the menu bar icon if you have it enabled.
The two entries in the notification settings are an unfortunate drawback of this system, as macOS doesn’t have a way to share settings between these two parts of the app. If you give both apps the same settings, then everything should work as expected.