Both Olympus E-M1 II and OM System OM-1 have Four Thirds sized 20.0 MP resolution sensors so sensor size and resolution is not a differentiator between these two cameras.
Another similarity between these two cameras is that both OM System OM-1 and Olympus E-M1 II sensors lack anti-alias (Low-Pass) filters. Removing anti-alias filter increases the sharpness and level of detail but at the same time, it increases the chance of moiré occurring in certain scenes.
Below you can see the E-M1 II and OM-1 sensor size comparison.
Olympus E-M1 II and OM System OM-1 have the same sensor sizes so they will provide same level of control over the depth of field when used with same focal length and aperture.