It worked without hesitation and immediately:
Bought 1 app via Fire TV, installed and started
2. On the Mac System Preferences / Displays called and tick "Show Sync Options when available in the menu bar" purely in
3. The fire was on TV Mac in the system menu bar "airplay" immediately detected and shown on TV Desktop immediately after activation of the Mac.
The rest was a little tweaking. But everything worked immediately without any problems as it should. A bargain for 4.- Eur compared to buying an Apple TV box.
The great thing: the app is in the Auto Load, ie always available and does not require any special start. Once an AirPlay client is active, the screen will automatically come to the fore. Convenient never go on fire TV.