Unfortunately not. It opens Chrome to authenticate with Google but never redirects back to the native app. So I 'installed' Home Assistant as a PWA and found there is practically no need for me to use the app. On iOS it does work with the native app thought.

Note that if you are not using Cloudflare Access as additional authentication layer and only rely on Home Assistant authentication, the Cloudflare tunnel obviously works with the Android app. It's just that I was too paranoid for this.

Maybe I'm overly cautious. Home Assistant does have two-factor as options as well, doesn't it?