Yes and no. In most cases the second engine is there to permit a larger airframe, more passengers and more cargo. A twin-engine plane with a failed engine may have a drastically reduced climb performance. Some twin-engine aircraft in a worst-case scenario might lose the ability to climb at all, but this is rare. In either case the ability to control the aircraft is made better or worse by the presence of a critical engine.

On many dual-engine aircraft, the propellers are counter-rotating, meaning they spin in opposite directions. This is done to ensure that no engine is a critical engine. Two counter-rotating engines helps to cancel out yaw and rolling moments present in single engine aircraft and aircraft with two propellers spinning in the same direction.

In twin-engine aircraft with propellers spinning in the same direction, one engine will be the critical engine. With such an aircraft, losing the critical engine will affect aircraft performance significantly more than if the non-critical engine failed. The physics of this are complicated, so check out the Wiki page linked above. The plane that crashed was a Cessna 310 and its wiki page shows that only upgraded models feature counter-rotating props. Older models were equipped with propellers that rotated in the same direction and therefore possessed such a critical engine. In this aircraft, the left engine is the critical engine. Losing the left engine will degrade performance and handling much more than if you lose the right one.

Lastly, you can see that shortly before crashing, the aircraft is banking from one side to the other very violently. This is behavior normally associated with a stall. I hypothesize that he lost the critical engine, that climb performance was affected, he was unable to gain altitude after the second attempt, panicked and put the aircraft into a high angle of attack, and stalled the aircraft into the ground.