Aura is currently built for iPhone. There is no separate iPad, Mac, Apple Watch, or Android app.
When you move to a new iPhone, sign in with the same account and restore your purchase if the subscription does not appear immediately. Your Wall of Wins photos stay on the old phone because they are stored on the device, not on Aura's servers.
Blocks only apply to the iPhone where you set them up. An iPad or Mac needs its own Screen Time settings.