Is the attorney-client relationship itself covered under attorney-client privilege or confidentiality?

I’m sure by now that you’ve heard about the revelation in court that longtime adviser and personal attorney to Donald Trump even before he became POTUS, Michael Cohen, also has as a client Fox “News” personality Sean Hannity. This revelation produced almost audible gasps across the political spectrum, and has caused quite a stir in the news/entertainment/infotainment industry.

I’m no fan of Hannity; in fact quite the opposite. The revelation led me to virtually howl with delight…at first. Then I started thinking about it.

The more I thought about it, the more I started to wonder if this is an issue concerning attorney-client privilege or attorney-client confidentiality.

Is this really a blow to the privileged communications or confidentiality, or is just the fact of an attorney-client relationship not covered by these principles?

Note: This Q has also been posted to that other board