So, I just dived into these 3D boxes and noticed the same thing! DualShock 4's right stick isn't recognized at all! This is almost three years later, and it still isn't fixed/implemented. Yeah, most people probably use just Xbox controllers, but many people prefer to use something else, and something else should work as well.
I think it has something to do with DS4 being a Dinput device instead of an Xinput device, and Launchbox/Big Box just doesn't see it as a dual analog stick controller. Every other button works just fine, just not the right analog stick. I know I could just use something like DS4Windows, but I don't want to because I don't need it anymore on my system for anything else! DS4 just works with almost everything just as it is these days, except rotating boxes in Big Box.
I guess I will just bind it to the left stick for now as a workaround, as someone else in this thread did. It's not a final solution, though.