Sound travels through water (hello Dolphins!) so compression/expansion is possible. I imagine shaking will create those kinds of waves and allow molecules to mix faster than if the water were still. As well as allowing floaty things like oil to move downward.
Key word, "almost." Taking the incompressibility of liquid too literally could mislead one to think that liquid in a sealed tube could be used for faster than light communication.
I imagine that much would depend on the shape of the bottle and the kinematics of the shaking. If it's a perfect cylinder, and the shaking is perfectly aligned with the axis of the cylinder, then perhaps no mixing would occur*. But if you shake a plastic soda bottle, the nubbins on the bottom would make areas of lower and higher pressure, which could induce turbulence. Also surface effects would be interesting -- either oil or water could be stickier, which could result in films, then droplets, being separated from one fluid into the other.
* though, shockwaves could conceivably form circular vortices...
There is no "not ... at all" in physics. Liquids have, strictly speaking, a very very low coefficient of compressibility, but are not absolutely incompressible.
