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
Like you, I couldn't get my head around the difference between the different types of App Service apps (Web Apps / Mobile Apps / API Apps). Judging by the fact that no one has really answered your question, it looks like we weren't the only ones.
It made no sense to me that Web Apps, Mobile Apps and API apps were distinct things: surely the whole point is that you should be able to expose an API that serves your web site, your mobile apps, any other kind of client, making use of whichever features you need for your use case?
And, lo and behold, buried deep in the Azure docs:
The only difference between the three app types (API, web, mobile) is the name and icon used for them in the Azure portal.
Yep: they're exactly the same.