Apple Music cannot connect to internet (download stops with error 7506, pages say offline)

Though I have full Internet access on my PC (Ethernet as well as Internet), I cannot play Apple Music titles, it just skips the Apple Music titles and plays the first title found in my local music library

When I play the same title in Apple Music Web player (same computer, just in the browser window), it works:

I have already checked all my Apple music settings, it is set to synchronize with icloud everywhere using the same Apple Account. And of course I have an Apple music Abo.

I had the same problem in itunes, before I installed Apple Music on my Windows 11 PC. After some synchronization time, the music library disappeared from itunes and was only visible in Apple Music app, I was hoping, that the problem disappears now, but same problem there.

I also disabled the Windows Defender firewall and checked, if Apple Music is listed in the allowed application, nothing helped.

Sometimes when I open Apple Music app, it even doesn't load my local library, and it says "You are offline", even though I can access the Internet using other apps on the same PC, see below screenshot: