Happened to me on my personal laptop at least for a few versions of macOS (not really sure because I don't use that computer so often and stopped paying attention to it). I noticed it started happening also on the current work computer, which I had for about 4 months now, so I tried again some combos to see what was the current (buggy) behavior.
On my machine at least, deleting ~/Library/Preferences/com.apple.Music.plist does the trick (expect to have some personal preference zapped, of course, and to see the Apple Music welcome window even if you are not required to sign in again):
quitting the app and reopening it results in the window visible and in the position you left it, like one would expect.
Except, if you send the app to full screen and quit the app from there, the bug is triggered again (reopening the app, the window is hidden).
Long story short
- delete ~/Library/Preferences/com.apple.Music.plist (and possibly ~/Library/Preferences/com.apple.Music.eq.plist)
- remember to exit full screen before quitting Music
- it's about time @apple fixed this (and a few hundreds little bugs that have been around for years now), isn't it?