For saving logins and passwords, did you read the Usernames and passwords are not saved support article and do your settings meet the requirements for not using Private Browsing mode?

As for syncing logins and passwords, perhaps you have a Master Password set on one of your devices, preventing them from syncing? See How do I choose what information to sync on Firefox? for more info.

Do you use a master password?

• https://support.mozilla.org/en-US/kb/use-master-password-protect-stored-logins

Try the Firefox version from the official Mozilla server if you currently use a version from the repositories of your Linux distribution.

• https://www.mozilla.org/en-US/firefox/all/

It seems to work now. I finally reinstalled firefox. Moreover, I created a new profile, and synchronized stuffs form this new one. Thanks for your help.