OK, I just twigged:
It doesn’t ask for a password while there are other IE windows open (except the first time).
Every time I tried it above, I also had this forum open in another window. If I then closed just the web admin, and reopened it again, it does not ask for a password.
If I close all IE windows, then open the Firefly web admin again, it asks for a password as usual.
I’ve just tried it with Google open in another window: Firefly asks for password the first time, but as long as another IE window is open, Firefly repeatedly logs in without one.
Not sure if that helps? (Or if it’s even a bug?)