Would a lighter aircraft require the same amount of power than a heavier one to fly at the same speed?
You are right. The heavier aircraft needs more power. With one exception.

The power needed to keep an aircraft in the air is the product of flight speed and drag. When flying at the same speed, the drag of both aircraft is crucial to answer the question. Since both share the same geometry, it is safe to assume that both have about the same friction and parasitic drag. But the heavier aircraft needs to fly at a higher lift coefficient in order to carry its higher mass, so it will create more induced drag.

The one exception is when both aircraft have a highly laminar airfoil with a pronounced drag bucket. If the flight speed is picked such that the heavier aircraft will still fly within this drag bucket but the lighter aircraft will fly at a lift coefficient below the lower boundary of that drag bucket, the difference in friction drag could be higher than the difference in induced drag (which is small at high speed). But that exception would need to be made clear in the question.

To fully judge the correctness of the test, please provide all answers. As @David K suggests, your error might have been not to select all correct answers.