Hey there Samibug24,
We'd like to help look into getting the songs from Apple Music to play on your Bluetooth Marshall speaker.
Thanks for the screenshots you sent, as we see your two Apple TVs show when going into your AirPlay settings from Control Center, but your Marshall doesn't show. To make sure, are you able to scroll down any more from the AirPlay settings. If yes, you'll want to check if your speaker is showing there.
You also mentioned updating recently, but we're unsure of what you meant as both a Bluetooth speaker and your iPhone could have an available update. While you'll need to reach out to Marshall regarding how to update the firmware inside of the speaker, we'd like to provide the steps for your iPhone 11. The latest version is iOS 14.4. You can verify the version number installed within Settings > General > About. Here's a link that'll show how to first back up, then Update your iPhone, iPad, or iPod touch.
You mentioned it plays your 'turntable' with no issues. If you meant the same speaker works with playing audio to a Bluetooth-compatible turntable, that shows it's working. Of course, this doesn't resolve the issue of it not showing within Control Center. We see in your first screenshot what looks to be an app not developed by Apple. If an app is required for your Marshall speaker to work with your iPhone, there may be an issue with the app itself. If this is the case, you may need to contact Marshall about this as well. To completely rule out being outdated, this link shows how to Update apps or use automatic downloads.
Another good idea would be to test another Bluetooth device, like another speaker, or even a headset. This will help narrow things down further.
If the issue remains after taking all of the steps above(including contacting Marshall), we recommend contacting Apple Support here: Get Support
Have a good one!