No, the 2 databases don't have to have the same "name". Just set the new server's name in the configure.php file. That would have already been done for you when you ran zc_install by following the FAQ I pointed you to earlier.
You're probably using different DB_PREFIX values on the different servers, thus causing the tables to not be found. The FAQ article also mentions this point as something to address during the move process. Maybe you missed that?




