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
Best Answer
When you open a new tab, it will not share any cookies (and thus login status) with any other tabs.
When the website opens a new tab, a connection between those tabs exists that allows Javascript to execute across the tabs. This connection is preserved in Private Tabs and also means cookies are shared.
I'm not aware of documentation for this, but it's easy to confirm. If you open GMail and right click to open a new tab, it does not stay logged in. But if you click a link Google configures to open in a new tab, it will stay logged in.