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.
Curtis missed the Apple TV Remote app which allows you to control your Apple TV Plus show or your Apple TV Channel on your Apple TV hardware device — including the Apple TV software running on the Apple TV device — but not Apple TV software running on non-Apple TV hardware. Got it?