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Both login keychain and iCloud Keychain are corrupted and showing pop ups , before following the articles make sure you have all your passwords for apple id , iCloud id , web mail , air port express / base station password if it is configured will be deleted and must be in your records

https://support.apple.com/en-in/guide/keychain-access/kyca2423/mac

https://support.apple.com/en-in/HT202755

Second method is open keychain access through spotlight , open its preferences and click on reset my default keychain .

Take views from apple care senior advisors see this link https://support.apple.com/en-in/HT201232

Passwords syncing in older Mac version and new Mac version seems to be impossible they must of same version .

See this article https://support.apple.com/en-in/guide/keychain-access/kyca1121/mac

Note: You can’t copy passwords stored in your iCloud keychain. To transfer these keychain items to another computer, set up iCloud keychain on the other computer using your iCloud username and password.

If you mean Mac book keychain password become unsynced ? with another Mac how? admin password is stored in user library > keychain > in the folder login.keychain_db

And admin password for both Macs are separate and the Macs have their individual system files and can't match other versions .

Restarting will never set up iCloud Keychain and login keychain , also it's not a bug , unnecessarily opening keychain access and locking the lock folder under login will create pop ups .

Also login keychain password is kept same as admin password , and see this article when passwords don't match the pops will then also occur https://support.apple.com/en-in/guide/keychain-access/kyca1242/mac

Uninstall I password , skype , third party applications might not be compatible with Mac ( consult about third party apps with senior care apple advisors ) , there is no such malware in system or hacking , your login keychain seems to be working the lock for login is always opened by default see this screen shot , for second hand Mac before using always erase the hard drive and reinstall Mac OS https://support.apple.com/en-in/HT204904

What was the status of second hand Mac when you started it , was the hard drive completely clean after that you created your own Apple ID and signed in see this https://support.apple.com/en-in/HT205749

And shttps://support.apple.com/en-in/guide/keychain-access/kyca2794/

And passwords are never hacked or read physically by others they are always encrypted and no one can hack your Mac when an admin account is created , automatic login is set off in users and groups , created tough passwords for all Apple ID , iCloud id , admin password or web mail password see this https://support.apple.com/en-in/guide/safari/ibrw1103/12.0/mac/10.14 also set up internet service provider router and its password , set up air port express see this https://support.apple.com/en-in/HT202068 ( see the section of media access control access ) .

Only the administrator can physically read them in keychain access see this https://support.apple.com/en-in/guide/keychain-access/kyca1085/mac

https://support.apple.com/en-in/guide/keychain-access/kyca1083/mac

Regarding third party applications for one password ( not required ) if you have set up iCloud Keychain the passwords are secured , the keychain folder are not deleted for keychain certificates see this https://support.apple.com/en-in/guide/keychain-access/kyca2794/mac

Skype - always avoid other applications , your system might get corrupted ( after all its your decision ) .

if you follow the second method resetting the keychain using keychain access preferences , all the passwords that are saved in keychain folder of user library will be deleted , and this keychain folder will be deactivated and a new keychain folder will be again created , you have to auto fill the passwords once again , this method is applied when both login keychain and iCloud Keychain are corrupted , not as separately for login keychain and iCloud Keychain , make sure you have written all passwords in a diary .

Note : enter in user library , click on finder > Go > hold option key > click on library > here you can see keychain folder .

The first method shows to reset iCloud Keychain , by signing in with same Apple ID and password both in Mac and iPhone on the same network and its quite complicated , if a single incorrect step is applied more complications can occur , that's the reason I suggested to consult apple support advisor on phone .

Your keychain password is same as your admin password the one you use to sign in Mac , it is remembered by the administrator , it can't be recovered but can be changed from recovery mode or in users and groups .

If an administrator has completely forgotten the password and he can't login in the Mac as he had checked the box of name and password in login items , and rest of all boxes are unchecked then see this method https://discussions.apple.com/docs/DOC-12712

And if the administrator knows the old password but wants to change the password from recovery mode only and don't want to use other methods see this .

• Close all applications and shut down the computer , press command , R and power key simultaneously , release only power key and keep on holding command and R key until apple logo appears , the bar starts loading let it be completed .
• Select the language as English , go to utilities on top menu bar > Utilities > Terminal execute a command resetpassword and click on enter key .
• Window appears , select your admin name it will prompt to enter Apple ID and password 

  Once you enter the credentials , two options will be shown ( a ) previous ( b ) next , click on next .

   Some fields will appear .

   New password ……….

   Verify password ……….

   Password hint ……….

   Enter your new password in two empty fields and keep password hint as empty ( optional ) 

   It will again prompt two options ( a ) previous ( b ) next 

   Click on next .

  Then it will prompt to restart the computer , so after restarting the machine .Pop - up will appear.

       The system was unable to unlock your login keychain , if you remember your old password you can update the keychain password .

      If you do not remember your old password , you can create a new login keychain or choose to leave the login keychain using a different password .

 Would you like to update the password , create a new keychain , or continue the login ?

  A window will appear on your desktop .

 ( a ) continue login .

 ( b) create new keychain .

 ( c ) update keychain password .

Click on update keychain password , then log out from your computer .

When you again login in the fields enter your “ admin name “ and “ old admin password “ , so that new password will be updated with login keychain .

Hence , the admin password is changed with the help of “ Recovery mode “ only .