The same issue is happening to 3 out of 5 of my family members. It seems to be a keychain service or server-side issue, as it doesn't not matter which device they use, all devices (including desktop Macs) signed-in with their Apple ID request to update Apple ID settings over and over. Example: I can take an iPhone or iPad that is signed-in as me (that does not have the issue), then reset/erase it and set it up as a new device for one of my family members, and when signed-in with their Apple ID that same device immediately starts prompting to update Apple ID settings. So this issue seems to be affecting certain Apple ID accounts (not specific devices).
The question I have is, how many customers are affected? From my small sample size of extended family members, it feels like at least 50% or more of Apple IDs have this issue (many may not even be aware they are having the issue if their settings app is in a folder and they don't pay attention to the red badge).
The functionality that is blocked is that you can't save or update passwords into your iCloud keychain and have it sync across your devices. If you add a new password to your keychain, that password is saved only to that one device and is not sync'd to other devices via iCloud Keychain.
My advice is don't spend any more hours on this. I've spent countless hours trying to fix this for my family members, many hours on the phone with apple support, changing passwords, erasing and resetting devices, etc. This is going to have to be escalated to engineering and either a software patch or a back-end server side patch will be needed to fix it.
One last point: In my home we have upgraded to the new HomeKit architecture (we use HomePods as hubs to run our smart devices, lights, etc.) I'm curious, are any others that have this issue also HomeKit users that have upgraded the Home app and HomeKit to the new architecture? It's probably not related, but I was just curious if perhaps there was a correlation between this iCloud Keychain issue and Apple IDs that are a member of an 'upgraded' HomeKit home?