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. Wall of Wins photos can return only if their earlier private uploads succeeded and you sign in to the same Aura account.
Blocks only apply to the iPhone where you set them up. An iPad or Mac needs its own Screen Time settings.