The iPhone and Android apps are the same CalmHandoff as the website, with the things a website can't do on its own: push notifications, opening files in your phone's viewer, native printing, and links that open in the app.
Sign in with the same email and password. There's nothing to move across — it's the same account and the same record.
You don't have to use the app. Everything works in a phone browser too, and some people prefer that on a shared device.