What influences prepreg thickness?
"Pre-preg" is a sheet of FR4 which is pre-impregnated with uncured epoxy resin. So it is flexible and somewhat plastic in nature.

Pre-preg is used in between pieces of "core". The core is, as you say, made with copper on both sides. The copper is then etched. Basically, core is like a two-sided PCB. Once all the layers are stacked together, the epoxy and glass fiber of the pre-preg fill the gaps between the copper areas in the core. Because of its plastic nature, the pre-preg oozes into the gaps. This is why the finished thickness of the pre-preg depends somewhat on the fill factor of the copper layers adjacent to it.

On a controlled impedance board, the manufacturer will manipulate several variables to achieve the desired impedance. This may even include slight changes to the trace width. I do not believe the control of pre-preg thickness is any different in a controlled impedance board. Controlled impedance is not magic. The layout still needs to be designed to achieve the target impedance. The manufacturer will just guarantee that it ends up within a certain range.