In legal phrasing, I'm thinking the attorney-client privilege applies to the client, and that a duty of confidentiality then applies to the attorney.
However, can it also be said that the attorney-client privilege applies to the attorney? Or does the attorney only have a duty of confidentiality and not a privilege?
I'm not sure what you are asking here. The attorney-client privilege and the duty of confidentiality are really two different things. Confidentiality is broader than the attorney-client privilege as the privilege only applies to communications made to the attorney by the client for the purpose of securing legal advice. The duty of confidentiality applies to any information obtained by the attorney (not just client communications) arising from the course of the representation, which can only be revealed with the consent of the client.
So, yes, the attorney is under a duty of confidentiality as Jacy noted. The attorney cannot reveal any information without the client's consent, whether obtained through communications (atty-client prvilege) or by other means (confidentiality) that might harm the client or is not impliedly authorized by the representation.
With the atty-client privilege, the atty. must claim the privilege on behalf of the client if the client is unavailable or cannot assert the privilege on their own.