You can use Alexa to control YouTube on Fire TV though: just say "Alexa, open YouTube" and away you go.
The official app also responds to other Alexa commands, in order to find, launch and play/pause/rewind and fast-forward videos. You can sign into your YouTube account and see your subscribed channels, playlists and recommendations. And, if you have the Fire TV Stick 4K, you can playback supported clips in 4K HDR and at 60fps.
The app is available for both of the most recent Fire TV Sticks now, with support for other Fire TV devices coming soon.
As for Amazon Prime Video, it is Chromecast-enabled at last. This means you can send a Chromecast or compatible device video via the smartphone app on iOS or Android.
You just need to download the latest update to the Amazon Prime Video app for your respective device.