Gift cards are not defined for a specific device.
They are redeemed onto an account, and can be used on any device that Apple ID is signed into including an Apple TV.
Once redeemed, the account balance should be used automatically when making a purchase, before charging any other payment method added to the Apple ID.
There is unfortunately no way to view the balance on an Apple TV though.
If there is an issue with other payment methods on the account, it will ask you to verify and resolve them, before letting you use your balance. But once its been corrected and the payment method method verified, it will use the account balance first before charging the other payment methods.