It doesn't use the push notification API, it uses the VOIP API which is briefly described
here.
I just did a test. I turned off push notifications, then logged into Skype. I exited Skype with the home button, and waited 45 minutes. I called the device and the phone rang. How would it do this if push notifications are turned off? The answer is that some part of Skype is running in the background, and not using push notifications.
The app the OP mentioned sounds like a good solution to the battery problem I get when Skype is logged in and accepting calls.