The "unix_socket" has been called by mysql authentication process (maybe related to a partial migration of database to mariadb, now removed). To get all stuff back working go su:
mysqld_safe --skip-grant-tables &
This will completely stop mysql, bypass user authentication (no password needed) and connect to mysql with user "root".
Now, in mysql console, go using mysql administrative db:
To reset root password to mynewpassword (change it at your wish), just to be sure of it:
update user set password=PASSWORD("mynewpassword") where User='root';
And this one will overwrite authentication method, remove the unix_socket request (and everything else), restoring a normal and working password method:
update user set plugin="mysql_native_password";
Exit mysql console:
Stop and start everything related to mysql:
kill -9 $(pgrep mysql)
Don't forget to
exit the su mode.
Now mySQL server is up and running. You can login it with root:
mysql -u root -p
or whatever you wish. Password usage is operative.