Why do "they" portray colliding black holes like that? When I see an animation of two colliding black holes anywhere, it looks like two merging beads of water. But why would the event horizon form a bridge during collision? There is no material flowing in-between. Wouldn't the event horizons "push" each-other away as they merge?(the effect not the black hole itself) For what I understand the gravity would poorly, but surely, be canceled out. Wouldn't this "push" only disappear from view as they got significantly closer, where it still exists but is trapped in a much larger one?

Just to make sure, as I see the similar questions: I don't mean what happens as they collide, but rather why isn't this portrayed if it does this effect at all?
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.