What is the relation between maximum airspeed and engine power?
In theory, yes. However, everything else isn't equal. 

When you double the speed, you probably exceed the design parameters of the airframe. You'll definitely exceed the VNE of a 100mph airframe if you operate it at 200mph, and then all sorts of ugly things can start happening, from weird handling characteristics to large pieces, like the tail, coming off inflight. Remember that 4x drag means 4x the strain on the airframe. Is your airframe up to the task? 

Most 100mph aircraft have a design optimized for short takeoff and landing, wings that are very efficient at slow speed, but not efficient at higher speeds. So the simple drag calculation may not hold true, when you take wings designed for slow speed and operate them at a much higher speed. The actual drag will probably be higher, if you consider the design of the airframe. 

Compare a 100mph aircraft, say a Cessna 150, to a Mooney 201, that can go 200mph. The Mooney has much thinner wings, that result in a high landing speed, but lower drag at higher speeds, while the 150 has a fatter wing that gives a lower landing speed and better low speed handling, but may not fare so well at a much higher speed than it was designed to handle. The other tradeoff, aside from speed, is the 150 is far more forgiving of a clumsy pilot, while the Mooney is probably not the best trainer for first time students. 

Note that the Cessna goes 100mph on 100hp, while the Mooney goes 200mph on 200hp, so the advantages of designing for the target speed can be seen in not just upping the power, but redesigning the airframe (and adding retractable gear).