As we can see you are using the WHMCS bridge. The WHMCS Bridge pulls in the WHMCS into WordPress. That’s why the search form design found from the WHMCS. There is no relation between the WHMCS search form design with our Hoskia theme search form design. Because both are two separate systems.
Moreover, we are sorry to say that there is no way to get the search results on the same page as you can see in our theme live demo.