Zwift on Apple TV

To my knowledge your subscription should carry over between different devices. I can Zwift on Pc and Android with the same subscription.

I found this on Zwiftinsider
If your Zwift subscription is managed through iTunes/Apple, resubscribing is easy.

Just go to Settings, tap on your name at the top, then click “Subscriptions”. Tap Zwift in the list to manage your subscription.