The title explains the question fairly well. The rear suspension also dips when power is applied (in the case of launch control). Does this phenomenon occur on independent suspensions, and why exactly does it happen?
It depends on the rear suspension of the vehicle as to whether it will "dip" or not under the circumstances you're talking about. The type of rear suspension I'd suspect you are describing is a cantilever design, where there is a forward pivot point at the body, then the two hub ends project towards the back of the car. This forms a large "U" shape if looked at from above. Springs support the car at the centerline of the wheel hubs and near the outside close to the hubs. When brakes are applied, the car can move slightly in either direction (forward or aft). When the car moves forward, the tires rotate and the axle moves with it (because the brakes have locked the wheel/tire stationary to the axle), rotating up towards the body, which compresses the support springs, thus making the back end dip. If you did the same in reverse, the axle would rotate away from the body, making the body push up.
Other types of suspensions (like independent rear suspension (IRS) or straight axle) don't normally do this because the axle itself cannot rotate (or in the case of the IRS, there's no axle there, but rather suspension parts much like what is in the front end of a front wheel drive car).
Here is the design I'm talking about in an image:
Part #6 is the axle I'm talking about.