I thought that it was not allowed to customize the taskbar on the iPhone. However, I noticed this app called iHandy Tip Calculator which replaces the battery icon with a fast switch icon, that pops up a view as shown. How is this allowed, and if so how can I be doing something different?

If you run the app on an iPad, you’ll see, how they do it: they are overlaying the normal status bar. Also note, that the screenshots in the app store don’t show this feature. I assume, they were fearing to get rejected and disabled the overlay for the screenshots.