Er no, your Mysql database for your Zen Cart installation is the culprit. Or, your backup of your original site on the other server is the culprit. If you look at the backup file from your original site, it is full of SQL statements such as

DROP TABLE IF EXISTS `address_book`;
CREATE TABLE `address_book` (
`address_book_id` int(11) NOT NULL auto_increment,

followed by a whole lot of other fields, and at the end something like

ENGINE=MyISAM AUTO_INCREMENT=24 DEFAULT CHARSET=latin1;

The above sets address_book_id as an auto incrementing field, and the last line says there are 24 entries already, so carry on from there

However, I usually SQLyog for such things, and instead, it creates

DROP TABLE IF EXISTS `address_book`;

CREATE TABLE `address_book` (
`address_book_id` int(11) NOT NULL,

in the backup file, omitting the auto increment.
When you attempt to add something, it fails because it is trying to use a key index of 0, which already exists.
To fix this, you have to edit the database, or edit the backup file, to add the auto increment where needed, which is just about everywhere.