I decided to try a workaround to test push notifications. Currently, I jailbroken a device and I'm using this specific device just for push notification testing. I wrote a tweak to create an extra routine on the publishBulletin method (method used to add notifications to the notification center) in the iOS core. Therefore, I get the needed data in that method and export it to a log. I might not be able to access the notification center data, but access an external text file is possible.
I know this solution is not the most elegant one and also has nothing to do with appium, but I hope this might help someone. In addition, with the device being jailbroken, I am able to change the iOS date and time using the terminal, which is also needed for most push notifications to be triggered.