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An account backs your data up off your phone and keeps it in sync everywhere you sign in. It's completely optional, and logging always works without one.
No. Every workout, meal, habit, and measurement saves to your phone whether or not you sign in. An account does two things: it backs your data up so a lost phone doesn't mean lost history, and it syncs your data to any other device you sign in on, like an iPad. Progress photos are the one exception. They stay on the device they were taken on, on purpose, and never sync.
Install Axis on the other device, go to Profile → Account, and sign in with the same email and password. Your workouts, meals, habits, and measurements download automatically. Anything you log while offline syncs on its own the next time that device is online.
On the Account screen you have two options. Sign Out keeps all your data on the phone. Sign Out & Erase This Device removes all Axis data and your saved API key from that phone, while anything already synced stays safe in your account. Use the erase option on a phone you're handing off or selling.