Once you enter the code on the screen accessed via the QR code or link and click the 'Sign in with Apple Music' button, a pop-up window should appear for you to log into your Apple Music account through Apple's system. After logging in, a success or error message will be displayed below the 'Sign in with Apple Music' button.
In your case, your browser might be blocking this pop-up window. I haven't encountered any issues during my tests on the application side. Additionally, please note that you must have an active Apple Music subscription to use this app.
Could you please check your pop-up settings and let me know the result? Specifically, do you see an error message appearing below the 'Sign in with Apple Music' button after logging in via the pop-up?