play music on iphone while apple tv is playing

I assume you're running current or fairly-current [i | tv]OS on all devices and you're referring to the media controls that appear on your lock screen (prior to opening Apple Music at any rate).

If you tap the little AirPlay icon, it'll open a list of what's playing where and nearby devices that could be used. At the bottom there's a "Control Other Speakers & TVs" button that lets you... do that. Poke around a bit further and you'll figure it out.

As for what's happening in Apple Music on your iPhone - can I assume you're playing something other than Apple Music on the TV? Nonetheless, I'm not a current AMusic subscriber so... I'll bow out.

So check out what's displayed when you tap the "Control other..." words at the bottom of your screenshot. It does look like just a big caption but it's a button. As I recall, it used to be a simple icon of a TV screen... but I guess folks didn't know what to do with it :-).

You're not seeing the iPhone in that list because it's not acting as an AirPlay speaker that's available to play the (currently controlled) TV's audio stream.

What happens when you just press "play" within Apple Music on your iphone?

Well, I see iPhone and "AirPods" at the top of my AirPlay list:

It looks as if you have so many AirPlay speaker options that your iPhone has been pushed up off the top of the screen. Can you scroll up? If not, could be a bug... try shutting down some macs and homepods to see if your iPhone reappears at the top of the shortened list.

Back to the local instance of the Apple Music app running on your iPhone - you're saying when you open the app and play something it affects your ATV? Or it plays on your TV Pods? Maybe you did an accidental handoff to the TV Pods?

Well, that's all pretty weird. :-).

Again, what happens when you start playing audio using an app on your iPhone?

What do you see when you shut down everything but your iPhone? When did you last restart your iPhone?