> The website/app instance is running on the phone/device/whatever. It is not passive. The user opened it ( or opted to keep it open in background ).

That's not how it works. Unless it's the foreground application, the application is not running; it's not kept open in the background. Instead, what happens is that the application registers with a push notification system when the user first opens it and logs in. When the user should be notified, the website sends a message to the push notification system's central servers, which notifies the phone (the push notification system does stay open in the background, but it's shared by all applications in the phone), which then starts the application on the phone to show the notification.