In order to host a different site under an additional domain you will have to use Cpanel's feature called Addon domain. This feature is available in higher hosting plans like Hosting PLUS, Semi Dedicated hosting, etc. If you are using the Standard shared hosting by SiteGround, but want to use the addon functionality you need to upgrade your hosting plan from the Order section of your Customer area.

Once you have the right plan allowing addon domain functionality, you need to set up your addon domain through the cPanel -> Addon Domains section in order to activate it. Click here for more information on how this can be done.

Finally you can easily install applications on your additional domain from your cPanel -> Fanatastico by selecting the addon domain from the drop-down menu on the application installation page.