I have this same issue in my S22 and newly bought S24. I'm not seeing the Spotify or Samsung Music icon in the status bar. But I do see it on my lock screen and notification screen. I do have my notifications allowed for the app.
That's the only explanation that makes sense "It's a feature, not a bug". Do you also not see it? I think it's so weird, why would the music apps not display icons in the status bar..