If the liquid didn't move, it would kind of magically find itself OUTSIDE of the container. So yes, when you shake a container (move it with strong if short accelerations), the liquid inside will too move accordingly, being subject of the acceleration provided by the container (thru compression waves).
Assuming that the container is closed from all sides, including the top, what happens also depends on if it is full, or if there is some space filled with air, some other gas, or vacuum.
If there is space, the container will move. It will also change the center of mass as the liquid splashes around.
If there is no space and the container is full of liquid, other answers describe what happens.
This is important when transporting liquids on boats. If the liquid can move around, the motion can flip the boat. Containers get designed to minimize the chance of that happening.
There is no space, unless you left a vacuum, and sealed the container under this vacuum. And even then, the liquid will evaporate to fill the empty space with gas, up to some pressure (that depends on the composition of the liquid and temperature).
That said, if there are thus more than one phase in this container, what happens is dictated by Archimedes. When one phase is less dense than the other, then it will be sujected to a force equivalent to the "weight" or acceleration of its volume filled by the more dense phase. That, plus the surface tension of the interphase surface will dictate how bubbly the phases will change position during your shaking acceleration. This can lead to the creation of lots of small bubbles, up to mixing an "emulsion" with the right surface tension.
short answer, it does, the molecules still have inertia, the rotational part of the shaking (assuming it's not a perfectly directional shake) will cause some parts of the liquid to have a higher velocity than others, this in effect means they have more movement energy, so they push others with less energy out of the way (while of course equalizing energy along the way), this causes the liquid mass to have vortexes and mix.
If you shaked perfectly along one axis, they'd probably only move a bit, and not really mix, depending on how compression able and uniform the liquid is.
That would also require the molecules to be arranged in regular crystal with an axis parallel to the direction of shaking and also maintain that arrangement which for a liquid would require the molecules to be perfectly still (absolute zero) since a material with vibrating molecules that stay in a crystalline arrangement would be otherwise called a solid.
Liquid not compressible does not mean at the molecular level. The model is one of 'springy balls'. Even in a solid rod if you hit one end, the wave propagates on the other side. A drop of food dye on a closed container with water would nevertheless diffuse or flow.
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.
Yes it will, basically for the same reason you can cause a swirl of a liquid that fills a square container, but can’t rotate a solid inside a square container: in liquids and gases, molecules aren’t tightly packed, and aren’t (strongly) connected to each other, so even small forces can make them move.
