A lot of the configuration settings for osTicket are indeed stored in the database (helpdesk_title aka site name, helpdesk_url, admin email, etc). So while you could have one database server, and multiple instances of osticket on other hosts connecting to it you would have to figure out a way to modify the source to set at least those three fields statically based on the site that was being called.
You would be able to do that by statically editing /include/class.config.php
At a bare minimum the functions you would need to change are: getTitle, getUrl, getBaseUrl, and getAdminEmail.