A cup of coffee is definitely a solution, in the sense that it is composed of dissolved solutes. The solvation occurs once water is introduced. While, there is always a limit to how much of a substance can be solvated in a given amount of solvent, with coffee you're likely to not reach the point of creating a super-saturated solution in normal conditions. If you were to take a pot of brewed coffee, and boil it, removing the solvent (water) you will likely reach a point as to where the solution is supersaturated, followed by precipitation of a crude extract.
Now is coffee an emulsion? I have heard that typical brewed coffee contains oils from the bean (likely oils that can be seen on darker roasts). If this is the case, then we could say that coffee is both a solution and an emulsion!
EDIT: I know this is an older question but I want to clarify some definitions:
Under IUPAC definition of an emulsion:
Note 4: An emulsion is termed an oil/water (o/w) emulsion if the
dispersed phase is an organic material and the continuous phase is
water or an aqueous solution and is termed water/oil (w/o) if the dispersed
phase is water or an aqueous solution and the continuous phase is an
organic liquid (an "oil").
So, when I say coffee is a solution, I'm referring to dissolved solids in the aqueous phase (an aqueous solution). When I say it's an emulsion, I'm referring to the "oils" of the coffee dispersed in the aqueous solution. By the above definition, coffee would qualify as an o/w emulsion composed with an aqueous solution.