Skype stuck on "Signing in" window
The following procedure seems to have solved the problem:


Sign out from Skype
Quit Skype and wait until all connections listed in nettop are cleared
Open Skype
Choose "Sign in with a different account" from the Microsoft ID login screen.
Try to sign in with Skype Name (but fail, since I can't do that anymore after I linked my Skype account to my MS account).
Go back and sign in with my Microsoft ID.


The "Signing in..." window was displayed for a few seconds, then the home window opened normally. Problem solved. I'm guessing steps 4-6 solved the issue, but I have no idea why.