Just a few days after my last post it now is official. Catalina 10.15.2 has re-instated the ability to control the Music.app on your Mac with the iTunes Remote app on your iPhone/iPad/Watch.
There is one thing which may at first sight seems to be a bit counter-intuitive when setting up the connection between your iDevice and the Music.app on your Mac.
Up until High Sierra and iTunes the easiest way to connect iTunes Remote app with your iTunes library was with using/activating Home Sharing. This does not work with the Music.app in Catalina (at least not when I tried it).
In order to establish a connection between the iDevice and the Music.app on the Mac I had to deactivate Home Sharing in the iTunes Remote app on my iDevice. You can do that under Settings (upper right corner when you open the iTunes Remote app) Also I had to delete all previously established connections to any iTunes Library. You can do that under Edit (upper left corner in iTunes remote app).
Once you have done the above two steps you will see a new Setup Screen in the iTunes Remote app prominently offering you "Connect with Home Sharing" or "Connect manually" in smaller letters (sorry, I don't have English as my default language setting - so the real options may read slightly different). Here I had to tap on "Connect manually". Once I did that my iDevice appeared in the side bar of the Music.app on my Mac with a little iTunes Remote logo and I received a four digit code on my iDevice which I had to enter on the Mac. Now the connection was established.
One more thought on this:
Thinking twice it seems logical that Home Sharing has to be deactivated and can not be used to establish a connection between the iTunes Remote app and the Music.app on the Mac. Home Sharing on the Mac used to be activated or deactivated within iTunes and as such is obviously closely linked to iTunes and iTunes is no longer available in Catalina. Now in Catalina the equivalent of Home Sharing is called Media Sharing and this is not controlled within the Music.app but within System Preferences in the section Sharing. In addition, the iTunes Remote app also still needs to work with older systems still running on pre-Catalina versions of macOS with iTunes. There you will most likely still need to use the Home Sharing option.
So the problem of this seemingly counter-intuitive behaviour is more that we were so used to one thing that we did not really thought about the consequences of the transition from iTunes to Music.app and from Home Sharing to Media Sharing. Apple certainly could have documented this way better. Also it would be nice to see an update of the iTunes Remote app which gives you a bit more guidance in the setup procedure.