the whole thing to the 'ring flash look' in portraits is getting the shadow halo around your subject (ie a thin ring of shadow on all sides around the subject). The only way to do this is to have the background very close to your subject (to the point where he or she is as close as possible to touching the background without actually touching it), so that you actually do catch their shadow on the background.
This is in direct contravention to everything you normally do for portrait photography, and as Big Mike said, is a very niche thing.
My studio has one booth where we sometimes do this look. Occasionally it really works well and can be truly stunning, especially on a model with strong features, but often it just looks weird.