Hi there,
If your branded app is provided by Moodle HQ, yes it should work with Google Analytics. But it's not something that happens out of the box when configuring it in your Moodle site, so make sure to contact us if you want that set up.
If you're building your own custom app, then it won't include that functionality because that is not part of the Free plan. You are of course free to implement that yourself by forking the open source repository with your customizations. You can learn more about the plans in apps.moodle.com.