When specifying the Site template mappings, custom templates are not displayed in the source site templates or destination site templates list.
A SharePoint site created with a custom site template doesn't hold a reference on the custom site template once the site's creation is completed. SharePoint only keeps a reference on the out-of-the-box site template (or WSP deployed site templates). As such, the ShareGate migration tool cannot retrieve the name of the actual custom site template that was used. Instead, the ShareGate migration tool will display the name of the out-of-the-box site template upon which the custom site template is built.
Although custom site templates won't be displayed, you can still use the name of the out-of-the-box site template upon which your custom site template is built for your mappings. For example, if you have a Custom Team Site template based on the Team Site site template, and you map the Team Site template at the source for the Blog site template at the destination, your Custom Team Site templates will be correctly assigned to the Blog site template at the destination after the copy.