How can I add buttons to the area highlighted in my screenshot? I don't see any mention of this in the extension docs, but I'm not sure what that part of the UI is called other than "title bar".
GitHub Copilot seems to be able to do it because it has an icon there in my screenshot, but I know that is maintained by MS and might do things that aren't normally possible though the extensions API...
There is no API available to that.
The GitHub Copilot you see there is built in on VS Code, not from GitHub Copilot extension. It was added to the recent launch of the free tier of GitHub Copilot, so the user has a kind of shortcut to install the extension.
Just as a side note, if you set workbench.editor.showTabs: none
on your settings, any menu added to editor/title
will be displayed in the title bar. Simply because there is no tab in the editor :-)