Music playing on Apple Watch but really on another device

My wife has this question about her Apple Watch but I see the same happening to me. I saw a similar question asked a year ago without a definitive answer. Resetting my watch is NOT an answer. I am looking for an explanation for what is going on. My wife's concern is that this is draining her watch's battery.

When I am playing music on my HomePod mini, we both see the music playing on our watches. The same is true if music is playing on my iPhone. It also appears. to be playing on my watch. Is this just a display of what is playing on my phone or on the HomePod? Or is music actually playing on both? When I test, stopping the music on my phone stops the music on my watch.

What appears to be happening is that the watch can control what plays on the other device. I just now stopped the music on my iPhone and from my watch selected the HomePod. The HomePod started playing what I told it. I do this all the time. If I don't want to yell for Siri to play something on the HomePod I simply select what I want from my iPad.

I also connected headphones to my watch and played music from the watch to the headphones. That looks different on the watch and is obviously using watch power to play the music via bluetooth.

So the question is, if the watch is showing what is playing on another device, is it draining the watch battery? It must be getting this knowledge from being on the same WiFi network with all these other devices. It must take some power to obtain and maintain that information.