Users of Fire TV devices (in the US) should be able to optionally subscribe to Apple TV+. Other purchases (buy/rent) or other subscriptions (in-app channels) in the Apple TV app need to have the transaction happen on an Apple Account from a compatible device that allows in-app purchases with Apple directly.
You can subscribe to Apple TV+ on the web, or on a Windows PC, or Roku, or PlayStation, or Xbox, or Apple device, LG TV, Samsung TV, Vizio TV, so that the subscription or purchase gets linked to your Apple Account. Once the transaction is done, that content will also be available on other devices, like Fire TV.
2. In the US, you can subscribe to Apple TV+ on Amazon Fire TV, but you can't purchase other content from the Apple TV app. On Amazon Fire TV (in other regions), Android TV, and Sony TV devices, you can't purchase content from the Apple TV app. You can buy, rent, or subscribe on another supported device and then watch in the Apple TV app on your Amazon Fire TV, Android TV, or Sony TV device.