Obviously having two perfectly identical and opposite vortices generated in exactly the same space is just an ideal mathematical exercise. I suppose it would something like having two billiard balls moving one against the other along the exactly same trajectory and at exactly the same speed... and that multiplied for all the air molecules!
Anyway something similar actually happens also in reality: with contra-rotating propellers. And luckily enough, this interesting report is just right for our purpose. It shows flow measurements behind naval contra-rotating propellers. Measurements are done in three steps: with propellers stopped, with only one propeller rotating and with both propellers contra-rotating. Unfortunately it doesn't deal with isolated propellers since also the stern of the ship is present in the model. So first of all we have to understand how the water flows due to the hull when the propeller are not rotating:
The (black small) arrows in the plot tell us in which direction the water is locally flowing while the background colours tell us how fast it is flowing (blue is slow, red is fast). The three big white arrows "connecting the dots" are my own work and show more or less the path of the water flow. Following them we see that what the water does in this case of not rotating propellers, is basically closing behind the hull: it goes up and then bend to rejoin in the centre of the picture just behind the propeller axis where the ship ends and then the flow just "dies" there. This picture of the hull should help in understanding this movement (C is the plane where the speeds are measured):
Now, what happens when only one propeller rotates?
We see that the propeller causes the water to rotate anticlockwise around it: the same three white lines of before now clearly bend around the propeller. This whole movement of the wake basically resembles the movement of the tip vortex at the wingtip as seen from behind.
Finally, in the next picture both propellers are in movement:
The three white lines are now again like in the first picture: the contra-rotation of the second propeller has more or less restored the symmetry of the water flow, cancelling the anticlockwise dragging of the first propeller. This image resembles indeed the "smiley face" in the first one: obviously here we have more red than blue (i.e. higher speeds) since the propellers are doing their job accelerating the water, but the anticlockwise rotation given by the first propeller is now more or less balanced by the second one.
Will two counter rotating vortices cancel eachother?
Yes, almost completely, even in reality.