Why do "they" portray colliding black holes like that?
The event horizon is not a physical structure. Rather it is the boundary of a region of spacetime from which no information can escape. If you were (unfortunate enough to be) between two black holes, you may be in a position in which there is no net force, and yet you could not travel from your location to the point distant from the two black holes, no matter how fast you were able to move. 

If you were in such a position, you would be inside the event horizon of the two black holes. 

Finding the event horizon means solving the Field equations of General relativity. Exact solutions are known in several situations: A single mass concentrated at a point, a mass with a charge and a rotating mass. If there are two masses, then the Field equations have no exact solution, but can be solved numerically (with a fast enough supercomputer) It is the results of these calculations that you are seeing.

So, the shape you see is the event horizon, The event horizon of two nearby black holes can be calculated by a computer. Since you may be trapped between two black holes the event horizon will be deformed towards another black hole's event horizon, giving the "teardrop-> bridge->merger" appearance of the final moments of two back holes.