Thanks for getting in touch @psypat. Site Kit when set up will register your site with a single domain name, while also creating OAuth credentials unique to your sites SALT keys. If you’re switching a WordPress site from a staging site to a production site, the best approach may be to reset Site Kit first, before migrating or changing site URL – from the staging URL to live URL. Without doing so, depending on how you’re migrating your site, the plugin may not always recognize a URL change, resulting in the errors you’re encountering.
What you may wish to do, if you’re using Site Kit on a staging environment, is to show data based on the production site using our Helper plugin. You’ll find more information on this here. This may not suit you if you happen to copy over/replace all files from your staging site to your production site. It may, however, be viable if the tool you’re using to push your latest updates from staging to live allow you to select what updates you wish to push (ie. only content, database changes, etc).
So to summarize, Site Kit set up is specific to a single site URL, which is registered for Search Console, Analytics and other data. If you’re switching URLs, it’s not uncommon to encounter errors if you didn’t reset Site Kit first, or if Site Kit doesn’t recognize a URL change (which is not uncommon based on all the different methods to change a site URL/migrate).
Let me know if you have any questions with the above.