Why does a liquid not rotate with the container?
Suppose your cup was full of a lump of stuff that had no friction. Then as you rotate your cup there is no force between the cup and the stuff, and so it would not rotate. In the case of coffee - the friction is from the viscosity of the coffee. But, as your cup pulls some of the coffee around with it right at the edge, the rest of the coffee slips on that layer, rather like if you slip on a patch of coffee on the floor. Eventually, if you keep rotating the coffee, say on a turn table, the small friction will rotate all the coffee. Then, if you stop rotating the cup, the coffee will keep rotating from its inertia, until the small amount of friction has brought it to a halt.
I would think inertia is the primary reason the liquid doesn't start moving immediately. There will be a point where, once the container is rotating fast enough, the liquid will start to rotate (vortex). Once inertia is overcome, the liquids rotation will accelerate and eventually approach the speed of the container. Has a bit in common with the photoelectric effect, there needs to be an initial energy input that overcomes inertia (analogous to the work function).