I am not sure why you re-registered on your new Echo. I have 5 Echos in different locations and I also am a subscriber to Amazon Unlimited. I can play on any device or all devices, using the multi-room function. You don't have to register a device to Unlimited. Once you register your echo under your account it is registered to your unlimited too. However, if you are playing music on one echo and then go to another echo and issue the command to play music, it will let you know that another device is already streaming, and if you want to stream from the device you are currently speaking.
I hope this helps.