Not quite. All two-body systems orbit a common barycenter, but the mass ratio of the Earth-Moon system is so lopsided that the barycenter is inside the Earth, not between them like with Pluto and Charon.

No, because absolute size is not what makes a moon a moon. Our Moon is a moon because it directly orbits a planet, not a star. Charon is massive enough relative to Pluto that the former does not directly orbit the latter, but instead they both orbit a common barycenter located between them, making them a binary planetary system.