The app is built with different technology than the web (it uses Ionic Framework). So it isn't possible to use anything from the Boost theme.
Everything in the app has been built "from scratch" (meaning that we haven't taken it from Moodle). And from a design point of view, this is a good thing because it allows us to make an experience tailored to mobile devices. But it comes with some trade-offs, for example that not all features in web are available in mobile (and not all features in mobile are available in web).
For the navigation drawer in particular, we already have something similar:
But I know it's not the same, that's why I was asking what do you like from the nav drawer that we're missing in the app.