The account is logged in on Safari, but I don't know the password.
Is there any possible way to save the password and get Safari to autofill the password when I log out and fill the username?
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No, there is no possibility to retrieve the password at this stage. The session state (i.e. the fact that "the account is logged") is stored by the browser in a cookie. The password used to login is not stored anywhere (unless the user explicitly permitted the password to be saved, which is not the case you asked about).
This is a basic security feature protecting anyone from having their password compromised if they left the device unattended for a moment. For the same reason you cannot change the password using Facebook account settings without providing the old one.