We are still getting this message constantly on every device since last week. I’ve logged out and in, turned devices off and on, and it keeps appearing. Relates to iCloud Keychain and it has gone circular. Any other ideas?
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Looks like the problem is probably keychain related. Try turning off keychain on all devices as well and just don't use the keychain service until the next update.

Here are a couple articles on the web that seem to be successful with the same issue. Give this one a try first. Well written.
Fix Keep Asking Update Apple ID Settings Issue on iPhone



Update Apple ID Settings On iPhone? Here's What It Means & What To Do