No Apple Id Section in MacOS system settings

Click your avatar in System Settings. The name of that panel is Apple ID. Click Sign-in & Security.

At this point, it seems clear that there is some problem specifically in your account.

The procedure that I am about to suggest should do no harm, but of course... make sure to have a BACKUP before anything.

We are going to try and "fix" your account. At best, this will fix it and at worst we will be back to where we started.

1) Make a new administrator account. In the following, and for simplicity, I will refer to your account as "roy" and the second account as "admin".

2) Log out from "roy" and log in to "admin"

3) Go to System Settings->Users & Groups, click the ⓘ next to "roy" and then click "Delete User ..."

4) You will be asked what to do with the home folder. Choose "don't change the home folder"

5) Click "Delete User"

6) In the Finder, press Command-Shift-H to go to the home folder of "admin", and then Command-UpArrow to go one level up. You will see your previous home folder, now called "roy (deleted)".

Change the name back to "roy".

7) Back in System Settings->Users & Groups, click "Add User" and choose the very same name "roy".

You will be warned that a home folder with that name already exists, and asked if you want to use it. You do.

8) Log out from "admin" and back into "roy" and test.

Try booting into safemode and then check it again. If there, login. Then reboot back to normal and test.

Start up your Mac in safe mode - Apple Support (English)

The solution that worked for me

Go To Settings

Go to Internet Accounts

Chose iCloud

Setup your iCloud account there and then reboot

That is very weird.

Try Safe Mode, as Randall suggested.

If that doesn't help, you probably should reinstall Sonoma over the existing system.

The Apple ID section of System Settings should appear at the very top, above Wi-Fi. This is strange.

Do try using a search as Randall suggested. What happens?

For comparison, this is what I get when I type Apple ID

into the search box:

What do you see if you type in appleid in the search bar of System Settings at the very top?

Is it hidden, if you try scrolling down in the left side?

it should be right above WIFI

Try Searching “Account”

The new screenshot shows "Family", which I believe was not there before.

"Family" is of course related to Apple ID...

What shows if you click "Family"?

You can use the new account, of course.

Anything related to iCloud is no problem.

You'll still have to get access to your files from the old account on the new one, and that can be done but you'll have to handle a bunch of permissions issues. I normally don't advise this, but if you know what you are doing it is perfectly feasible.