Apple TV free year not working

You are receiving Apple TV Plus for a year, which is their streaming service for original content. The TV app was available long before that and not only houses all content from the iTunes store , but other Apple TV Channel subscriptions. Those are completely separate and have their own associated fees.

The offer for 12 months is for those who purchase a new qualifying device (Apple TV, Mac, iPod Touch, iPhone, iPad) from Apple or an authorized reseller. The user would then need to setup that device and claim the offer there. The offer is valid for 3 month after device activation. Refurbished or open box items from other sellers would not count and those who previously subscribed would not qualify.

If you believe you qualify, are within the claim period, and are unable to claim then you can contact Apple support

Apple TV Plus is for Apple originals only, it will not give access to content from the iTunes store (which is done via rental or purchase).

The service currently consists of 22 shows and 4 movies, with more added each month. You can see exactly what is included by going to 'Available Apple TV Channels' and selecting the Apple TV Plus icon, which is where you would have originally gone to claim the offer.

Correct! If you are just browsing the app then you will be seeing shows from the iTunes store, which require purchase, along with movies for rental or purchase. The app also has other Apple TV Channels, and they would be separate subscriptions, which have their own trials and associated fees thereafter.

Apple TV Plus is clearly advertised as a service for Apple originals, and the offer you claimed is for that service. There is no subscription service currently available that offers access to a full digital marketplace (i.e. iTunes) as well as giving access to other content providers. In fact, Amazon works very similarly, with their app housing their Prime video streaming service, Amazon Prime Channels, and their Prime digital storefront.