I would like to know what effect does an object have in the rigid body simulations, when the volume x density of the object doesn't match the mass. In most cases the default density is 1.0 in blender!
It's not that the mass doesn't match
density * volume, it's that the density is defined by
mass / volume.
Needless to say, the default mass of 1 gram makes most objects very much the opposite of "dense" ;)
To calculate physically accurate values for mass based on measured density of real-world materials, try using the Calculate Mass operator in 3D view > Tool Shelf (T) > Physics > Rigid Body Tools: