Apple TV is a hardware device.
Apple TV is an app on Apple TV that curates content you can buy
from Apple and also content you can stream through other installed
apps (but not all apps, and there is no way to tell which ones).
Apple TV is an app on iOS/iPadOS devices that operates similarly
to Apple TV on Apple TV. Apple TV on iOS/iPadOS syncs playback and
watch history with Apple TV on Apple TV, but only if the
iOS/iPadOS device has the same apps installed as the Apple TV —
and not all apps are available on all platforms. Apple TV is also
an app on macOS, but it does not show content that can only be
streamed from external apps on an Apple TV or iOS/iPadOS device.
When you spell it all out, as Curtis does here, it really does expose how confusing a lot of this is.