Authentication requires the opposite of privacy. If you don't want to be identified, you can't restrict anything to your identity.
If I'm authenticating with server A. I shouldn't have to carry ephemera from server B. A can interact with B on its own if necessary.
Bubbling up these architectural details to the front end is a symptom of the webdev cargo cult coming up with broken ideas that get fossilized as the status quo.
It kind of depends. See Kagi Privacy Pass ("Allows you to use Kagi Search with Privacy Pass, which cryptographically ensures that Kagi cannot tie that request to an account and allows for further privacy and anonymity."): https://help.kagi.com/kagi/privacy/privacy-pass.html