Hi,
that's really curious now
The zm_database_init script relays on the .my.cnf configuration because it is only a wrapper script which calls the mysql command. The script should work when you can just enter "mysql" without any options (no user or password specified) and you get the mysql command prompt.
So far the theory
One thing which is also possible is that you've mistakenly entered the system root password and not the mysql root password to .my.cnf file. That would result in the described error.
You can also have a look at /etc/my.cnf which contains the global mysql configuration. You can check if it differs form original configuration by running rpm -q mysql --verify which should give no changes (empty result) when nothing was changed.
You see I'm currently running out of ideas what could cause this error
Hope this helps