I have this problem or one very similar. I have two Macs on Big Sur, a MacBook and Mac Mini. The MacBook has been on Big Sur since November ands been signed into iCloud fine. The Mac Mini just got upgraded a little over a week ago, and I also just upgraded it to 11.1 last night...but I think it's never been completely signed in... since I noticed this problem it's been half-signed in... it has my account, some things are working, but it says "Update Apple ID Settings" and "Some account services will not be available until you sign in again." (Specifically, I updated Server.app and am trying to get a new APNS cert and that's failing and I think this is why.) I can sign out, then sign in again, but it succeeds but comes back to the same message above, I can't ever sign in fully.
I started looking into the logs, and found this, which is probably indicative of the problem, but I'm not sure where to go from here... obviously something's failing, but the "(null)" looks like it should not be there:
default 09:55:37.674020-0500 akd Saving account <redacted>@icloud.com (<redacted GUID>)…
default 09:55:37.674040-0500 akd Attempting to perform block in persona: (null)
default 09:55:37.674055-0500 akd No persona ID on account proceeding without setting context.
default 09:55:37.674078-0500 akd Performing block within persona Invalid
default 09:55:37.674222-0500 akd Performed block within persona Invalid
default 09:55:37.674248-0500 akd Saving of <redacted>@icloud.com (<redacted GUID>) succeeed!
default 09:55:37.674314-0500 akd Successfully checked in with IDMS.
(Somewhat amusingly, the three "e"s in "succeeed!" there are in the log, not my own typo, I checked!)
Another interesting side effect, when I sign out of iCloud on the Mini, my MacBook disables iCloud Keychain and I have to re-enable it... and, the first time this happened I got a message (on the MacBook) about iCloud now using end-to-end encryption and that it will use my password instead of my iCloud security code (perhaps relevant, I never had an iCloud security code for iCloud Keychain, I always used "approve on another device").
I have a feeling this change to encryption mechanisms is also related to my problem, but i have no idea when this changed... with Big Sur 11.0.1? With 11.1? Before then?
I'm going to upgrade the MacBook to 11.1 now and see if that helps anything, but that system is working fine!