When a new project is created from a template, the new project is also set to be a template.
If the Project Create View on the Project Object includes the Template checkbox field, this issue will occur. The Template (yes/no) value will be copied over from the template to the new project. In order to prevent the new project from being created as a template, the Template checkbox needs to be removed from the Create View. It can remain on the Edit view, if already there.