Yes, C dorian mode belongs to a same Bb major/G natural minor scale with a different starting (root) note. And, yes, we could say the whole thing is in G minor.

Each of the modes in the scale: Bb ionian (major), C dorian, D phrygian, Eb lydian, F mixolydian, G aeolian (minor) and A locrian have the same notes. All the modes have different fingering patterns, and they have a characteristic sound, as the half step intervals takes place on different positions on the scale.