What happens when you shake a container filled with liquid?

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.

Could get the results at home with a see through dish filled with cooking oil, embedded with tiny flakes of aluminium foil.

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.

A paint mixer for instance mostly relies on rotational inertia https://www.youtube.com/shorts/bSUbEzxGjAM

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.

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.

What’s more, the liquid will move even if you don’t shake it. See https://en.wikipedia.org/wiki/Molecular_diffusion, or https://en.wikipedia.org/wiki/Brownian_motion