I have created an Azure Web App today and it has a Mobile section in the Settings which contains Push notification, Mobile Authentication, etc. You can even download the source
Sorry for the late reply. They all are really the same thing. If you look in Azure Resource Explorer (available in the application menu of your app or resources.azure.com) you will see the indication of how we display a different icon for it - "kind": "mobileapp", for Mobile Apps for example. We leverage the different kinds to provide different quick-starts. You can certainly create an Azure Web App and load Azure Web App code to it to get the Mobile Functionality.