I'm developing an Orchard Core site and would like to place the shape templates into some sort of folder structure, if possible?
See example from tutorial:
So would it be possible to place the ColorField shape templates into a ColorField folder and the PersonPart shape templates into a PersonPart folder?

Not a direct answer to your question, but I was having problems getting Razor pages in a ~/Views/Shared folder to display as a shape.
JTKech replied here to me https://github.com/OrchardCMS/OrchardCore/issues/6376?notification_referrer_id=MDE4Ok5vdGlmaWNhdGlvblRocmVhZDkyNDcxMjQ2NDo3MjcwNjE%3D#issuecomment-640995088. Basically, OC looks at certain folders ("Views", "Views/Items", "Views/Parts", "Views/Fields", "Views/Elements") to display shapes.
That being said, there was a way in O1 to change what folders were searched, and I believe there is such a way in OC but haven't been able to find it.
In the end, you may want to place Color Field and PersonPart into ~/Views/Parts until you find a better solution to suit your needs.
Another option is to create a Theme and place those files within a similar structure there. I tend to do this with front-end views, keeping the Admin/Edit pages in my module. But there's nothing to stop you from doing them either way. Good luck.