The free year of Apple TV+ is tied to your Apple ID. As long as you are logged into all your devices with the same Apple ID you can access the content in the TV+ Channel.
Keep in mind that the Apple TV app and the Apple TV+ Channel are not synonymous. A subscription to the TV+ Channel does not get you inclusive access to everything you see in the TV app.
Apple TV, Apple TV app & Apple TV+
The Apple TV is a physical device that you connect to your TV with an HDMI cable and install apps on it, such as Disney+, Netflix, HBO Max, Fox Sports, etc. It also has the Apple TV app installed on it.
The TV app is an aggregator app that pulls in content from various providers, like those mentioned above, and is a one-stop-shop for keeping up with what you are watching and discovering new content to enjoy. Through the TV app you can also subscribe to various Channels such CBS All Access, Showtime, AMC+ and Apple TV+. There is no subscription to use the TV app.
Apple TV+ requires a subscription to view the majority of the content and it is billed through your iTunes account. A subscription to TV+ does NOT give you access to everything you can find in the TV app; only to the TV+ content.