Thanks for reaching out. It sounds that during the setup flow when requested not all permissions that are required to connect Site Kit with your Google account were granted. This isn’t a problem however, as you can redo the plugin setup. Please try this and then grant all requested permissions. Let me know how that goes.
in other side it has informed that; “Site Kit has already been configured by another admin of this site” while the account is same as before.
Site Kit can be setup by other WordPress administrators. Are there other WordPress administrator users or is it possible that you have a shared login with others?
In the case of another admin setting up the plugin this isn’t a problem. They’ll just need to ensure they grant you permissions to the connected properties if any.
In the case of a shared login other users may have setup the plugin, and this could be the cause of the permissions issue above. Ideally each user would have their own WordPress administrator account, and this would be associated with their own Google account so they can setup Site Kit themselves with their own Google account.
Please inform me if you have any questions with the above, or let me know how you get on after attempting setup once more.