When an organization assigns someone to create a site, make sure that they share edit or owner access to someone in the organization. It might be a good idea to have them sign a statement to that affect. There have been many cases where the site developer leaves the organization and refuses to give site ownership to anyone else.
You can add editors to your site.
While in edit mode click the 'Share with others' icon.
In the Share with people and groups window click the settings (gear) icon and make sure 'Editors can publish, change permissions, and add new people ' is checked. See more https://sites.google.com/pjrprojects.co.uk/newsitesguide/publish-permissions
After adding an editor you can transfer ownership to that account. Click the drop-down after the editor's name and change the setting to 'Is owner'. Your owner account will be changed to 'Can edit'. The site files will appear in the new owner's drive. The new owner can remove the original owner or leave him as editor. The only way to get ownership back is for the new owner to make you the owner. Before you transfer ownership you may want to make a copy the site so that you do not lose your work.
If your site is in Google Workspace and the other user is outside of your Google Workspace you cannot make them owner of your site. Your quickest option is to add them as an editor and then they can copy the site and they will become the owner of the copied site. This will have a new URL etc. (From a post by 'Steegle').
In the link sharing window make sure to select "Off - specific people". If you select :On - public on the web" ANYONE will be able to edit your site. See the image at the bottom-left of this page.