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

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you

You can check that you aren't clearing important cookies.

• using "Delete cookies and site data when Firefox is closed" to clear cookies keeps cookies with an allow exception
  in 102+ version toggling this setting makes changes to the "Clear history when Firefox closes" settings and those settings prevail
• using "Clear history when Firefox closes" in Firefox 102+ honors exceptions and keeps cookies with an allow exception, previous versions removed all cookies

Make sure to keep the "Site settings".

• clearing "Site settings" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions"
• Settings -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings

sorry, I don't really understand the way you said

When you close Browser or restart your computer/devices your login will logout of sites regardless that is for ALL Browser for Security reasons. So unless you have AutoSave/Fill the next time you go to the same URL they will fill in the login. That's only other way to save your the correct login to sites.

https://support.mozilla.org/en-US/kb/automatically-fill-your-address-web-forms https://support.mozilla.org/en-US/questions/1381263 https://support.mozilla.org/en-US/kb/password-manager-remember-delete-edit-logins https://support.mozilla.org/en-US/kb/where-are-my-logins-stored https://support.mozilla.org/en-US/kb/enhanced-tracking-protection-firefox-desktop