If I remember correctly, signing in to the Amazon Prime Video app requires confirmation through your account on the Amazon website or mobile app. Each device has to be registered with them. Be signed in on their site, scan the QR or visit the given URL, copy the code if applicable, and you’re done.
A regular Prime Video account can be used on three devices simultaneously, but you may not stream the same video to multiple devices. There is a limit for simultaneous use, but I don’t think a device limit per se. I am signed in to the app on all my compatible devices (5 devices).
The Prime Video app on Apple devices can use subscriptions through Apple, or Prime subscriptions through Amazon, either one will work, but they are not interchangeable. To change, cancel and re-subscribe the other way.