Could you please expand in what passwords you are talking about and what they are for as not sure if just Firefox or things like Sync or.... Please expand on what you are being asked to sign into. Thanks.

Fyi: If using Sync with out Syncing to another device you may as well disable all of that as it is not intended as a backup and you will loose your data.

A detail like a website remembering you (log you in automatically) is stored in a cookie.

You can use these steps to make a website recognize and remember you:

• you can create an "Allow" exception to keep specific cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Options/Preferences -> Privacy & Security
  Cookies and Site Data: Exceptions
• make sure to specify the correct protocol

In case you use "Clear history when Firefox closes" or otherwise clear history.

• do not clear the Cookies
• do not clear the Site Preferences

• Options/Preferences -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an "Allow" exception you may want to keep