On the native banner, it will show the native app logo, which is the Discourse one, because it’s pointing to this apps:
If you have a white-label Discourse app you will want to customize this lines:
Android:
https://github.com/discourse/discourse/blob/master/app/views/layouts/application.html.erb#L51
iOS:
That said, using the standard Add to HomeScreen on Android will use your custom logo everywhere, just set a good one at the Site Setting large_icon_url with 512x512 pixels.