We can put this another way by seeing that the surface has no thickness - it is just the places where the coffee cup ends and the air or coffee begins. Surfaces can be flat, like a table top, or curved like the surface of a football, a balloon or a soap bubble. The surface of water can be either flat without ripples, or curved when it has ripples or waves on it. We use the word membrane to mean a sheet-like 2 dimensional object, an object with area but very little or no thickness.
Good examples are sheets of paper or a piece of plastic food wrap.