Thanks for the question. We offer a support service to clone one site to another site but it needs to be performed by a Shotgun support team member; there isn’t a way to do this via the API. This is a full clone and you can’t cherry pick the site preferences/projects you want to copy.
Out of the box, you have the ability to copy Project Configurations from one project to another. This includes:
Project Navigation Settings
Record Detail Pages
Custom Pages
Tracking Settings
For more on that, you can visit the documentation here: