This question asks for help using the Safari browser and is not primarily about the Stack Exchange website, though that's part of the context.
When I use Safari in macOS for my Stack Exchange activities which often involve a dozen tabs or more, I'm frustrated that each new window I open by right-clicking on a link to an SE post within an SE post opens a new tag that's not logged in to SE, and I have to re-login again in each new window.
Curiously if I simply click on a linked question in a chat room that I'm already logged into (e.g. this one), it opens a new tab that is already logged in so I know this is possible.
Question: Is there anything I can do within Safari on mac OS so that right-click opening a link within an SE post to another SE post will open the tab as logged-in?
note: I regularly clear my cookies and decline cookies when I login to SE, and I always use Private Window browsing in Safari. I have Safari 14.0, macOS 10.15.7