Why does HN ask my browser for the PW even if I don't click on “Login”?

If you were not trying to upvote anything, then go into the web developer tools and select the console. Repeat the process that triggered a prompt. You should see what URL is triggering an authentication request. If that does not make it obvious, close the browser and start it in safe mode. Each browser [1][2] has it's own method of starting in safe mode to avoid loading any addons or custom code.

[1] - https://support.mozilla.org/en-US/kb/troubleshoot-firefox-is... [Firefox]

[2] - https://superuser.com/questions/254579/how-to-open-google-ch... [Chrome]

Did you click something else that requires being logged in, like "reply" or the upvote arrow (among other things)?

Any questionable extensions?