Can drone with two propellers above one another fly vertically upward without spinning?
Yes, because spinning in the opposite direction produces counterbalancing torque. Those rotor over rotor models adjust yaw by small variations in the speed of the rotors, so that one produces a bit more torque than another. 

There are a number of large helicopters that use two main rotors to the same effect, such as the CH47 Chinook (front and rear) and the Kamov KA31 which uses the same rotor over rotor configuration as the model you picture.