Constant pop-up saying Sign in with your Apple ID

First, check for any software updates. If there are no updates, try signing out of your Apple Id, then back in System Settings. Lastly, if the issue persists, there could be an issue with they Keychain entry for you Apple Id password. Check out this link to learn more about Keychain Access. Keychain Access User Guide for Mac – Apple Support (MY)

The issue was finally resolved by logging out of the iCloud account. Clearing and creating a new session on the iCloud account seemed to reset whatever was causing the problem.

I had a similar problem arise after upgrading to macOS 15.1 from 14.7.x.

In my case I had two "Apple Account Suggestions" appear in "System Settings" which would not clear after multiple attempts at clicking on them and entering the system password. Instead of clearing they would constantly reappear at the bottom of the list. The "Suggestions" are titled "Confirm Mac Password" and "Security Settings Verification Required".

At first I was concerned that the system and apps (Notes, Contacts, Photos etc) sync to iCloud was not functioning but I was latter able to confirm that all was well. My conclusion was that the system was functioning correctly and that the problem was simply a non-critical one of the notifications failing to be cleared.

I contacted Apple Support who suggested resetting the system password and then the iCloud password in attempts to resolve the issue but neither approach caused the "Suggestions" to disappear. The final recommendation was to reinstall the system which I have not yet been able to attempt due to other demands so I cannot confirm if a re-installation will resolve the problem.

Never seen this before

Start now and open an Apple Support Ticket as they are PAID  Apple Employees to deal will these types of issues 

1 800 MyApple

Other location Call Centres 

 Contact Apple for support and service

I think you will find that this selection issue is covered in the following Apple Support article:

How to reinstall macOS – Apple Support (UK)

Does the User ( you ) have more than 1 Apple Account ?

Which is possible

Copy and Paste from well respected contributor @Phil0124

" You can have different Apple IDs for each of the different services that use one: FaceTime, Messages, iCloud and iTunes & App Stores, but you cannot have multiple Apple IDs for the same service. Only one Apple ID can be logged in to each service at any one time. "

Indeed, I too saw the mention of the HD - Data disk, but it also says that if you are offered that choice, then select the HD disk:

"If the installer offers the choice between installing on ‘Macintosh HD’ or ‘Macintosh HD – Data’, choose Macintosh HD"

As an update on this issue; I followed Apple Support's recommendation of reinstalling the system to clear the bogus alerts however this did NOT resolve the issue and I am still experiencing the same problem.

I went throuh that process to no avail. See my comment below.

Unfortunately not; occasionally I will still get the popup screen. I still go through the process of entering the password just in case there is an issue.