Identity Crisis

Thank god, my interest in astronomy was around the time when we still had moons circulating planets and planets circulating stars.

Now, new planets are being added to the original list according to new definition, Ceres (earlier an asteroid), Xena (an object, whatever that may be) and Charon (Pluto's moon)

Does that mean that a planet (Charon) circles around a planet (Pluto)? Or does that mean we have a binary planet system (similar to binary stars) in our solar system? Crikey, those two must be facing quite an identity crisis...