I wonder if the capacitance is the same if we have two plates with a dielectric material between. I mean, is it the same when the dielectric material only touches one of the plates, compared with when it touches both plates?
No it isn't the same. After all, the electric field in the dielectric will be different (reduced). By placing an infinitesimally small metallic slab along the bottom face of the dielectric, you can model the capacitor as two capacitors in series, like in this diagram:
Hope this helps.