Are you using Sync?

Start Firefox in Troubleshoot Mode to check if one of the extensions ("3-bar" menu button or Tools -> Add-ons -> Extensions) or if hardware acceleration or if userChrome.css/userContent.css is causing the problem.

• switch to the Default System theme: "3-bar" menu button or Tools -> Add-ons -> Themes
• do NOT click "Refresh Firefox" in the Troubleshoot Mode start window

• https://support.mozilla.org/en-US/kb/diagnose-firefox-issues-using-troubleshoot-mode
• https://support.mozilla.org/en-US/kb/troubleshoot-extensions-themes-to-fix-problems

Thanks for the suggestion, cor-el. I haven't had time to try Troubleshoot Mode yet.

I am using Sync.

Strangely, since I posted this message, the problem has become less frequent. Perhaps this is because I did Help-> More troubleshooting -> Clear startup cache.

I think I've discovered a pattern in what provokes this problem: I clear site cookies and then refresh a page (ctrl-F5), and shortly after (not necessarily immediately) I get the prompt.

Are you canceling the PP prompt because that will (re)lock the passwords and prevent access?

You can temporarily disable the Password Manager by removing the checkmark on "Ask to save Logins..." to see if that has effect. You can still use the right-click context menu to fill a login in this case, assuming that you have entered the PP.

You can inspect the current login state and login and logout and change the Primary Password via the Security Device Manager.

• Settings -> Privacy & Security
  Security -> Certificates -> Security Devices

In the left panel select "Software Security Device". You can find "Log In" and "Log Out" and "Change Password" in the very right panel. You may have to Zoom out the page or go full screen to make this right panel visible.