The problem has a far easier solution than what I first looked for:
In the login screen, open the Accessibility menu – one of the icons in the top right corner – and disable High Contrast mode. So it's not a bug, it's accidentally for example hitting the shortcut key to enable the High Contrast mode.
It's made more confusing by the fact/bug that it's not really well working as a high contrast mode since the mode does not fit well to the Ubuntu's / unity-greeter's default background. Another confusing element is the disappearance of the user background, which feels like a bug while actually it's probably designed to force correct high contrast text if some original Lightdm default background would be used instead of the Ubuntu background.