The site ownership is assigned when one authenticates with Google credentials.
If no name is listed here then the site has not been claimed.
It can be claimed by clicking on the lock.
If the site is your site, your name will be displayed.
**If the lock is closed**, then you need to click on the lock and renew your authentication; otherwise, you will not be able to add or modify the content.
If someone else's name is here, then it is their site, not yours. You can read and temporarily rearrange their content--it will not change on their site. You can make a copy of their content if you have your own site to copy it into.