Hi,
Yes, that's precisely the problem. You're missing the primary key and auto_increment on the table.
For some reason, it seems your migration removed it...
You can run this MySQL query via your PHPMyAdmin to add it back:
ALTER TABLE #__hikashop_email_log MODIFY email_log_id INT(10) AUTO_INCREMENT PRIMARY KEY;
However, this query will only work after you've deleted all the elements with the same email_log_id, and since you have many entries with the value 0 in it, you'll have to delete these first.
A simpler simpler would be to just delete the hikashop_email_log table completely from the database. HikaShop will automatically recreate it next time it sends an email with the proper primary key and auto_increment.