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I figure the problem out.
Drush db connection is superseded by the .my.cnf located in the /root folder that allows the root user direct connection to mysql.
However since drush tries to connect to drupal@localhost that has a different password, it fails with Access denied for user ‘drupal’@‘localhost’.
Moving .my.cnf out of the way solves the problem.
It seems to me, better to add line
user=root
into the /root/.my.cnf file instead of renaming it. This will allow you to be able both update drupal and login to mysql cli without typing password.