You have signed out completely the iCloud account from system preferences > iCloud , it depends upon you again to sign with Apple ID say created with 2FA , or another id say example1@icloud.com or an Apple ID say with example2@gmail.com or with other mail services and these id don't match / similar with the primary Apple ID .
For that you need a iCloud Keychain verification code , if you don't have this code in records an iPhone is needed to set up iCloud Keychain .
So , what to do ? you must have a trusted iPhone to set up iCloud Keychain or code in previous records , you have only one option to use iPhone .
See this link https://discussions.apple.com/docs/DOC-13681
These points are important .
( f ) In system preferences > iCloud the keychain box will get unchecked , so tap the keychain box a pop up window will appear to set up iCloud Keychain on this Mac , request approval from one of your other devices using iCloud Keychain .
( g ) Three options will be shown ( 1 ) use code ( 2 ) cancel ( 3 ) request approval .
If you click on use code - this can be filled up if you had in previous records .
If you click on request approval - need an iPhone .
So without adopting any one method iCloud Keychain can't be set up , so the solution is arrange an iPhone .
In this article also https://support.apple.com/en-in/HT202861
See that both iPhone and Mac are simultaneously compulsory required to set up iCloud Keychain , its your choice to select any one .
- Get a Random Security Code.
- Don't Create Security Code. If you don't create an iCloud Security Code, then your iCloud Keychain is stored locally on your Mac instead of Apple's servers, and will update only across your approved devices.
If you create a random security code - the Mac will generate at random code that could be alphanumeric .
And if you don't create a security code the passwords will be stored locally in keychain and not on iCloud servers .
So, the basic theme is need an iPhone to set up code / or some Mac users create the codes as I mentioned in the above post and utilise it for settings up iCloud keychain .