Is a cup a unit?
Yes. In a US recipe, it means the customary cup. The US customary system of measurement includes a unit of liquid volume, the US gallon. One gallon is 231 cubic inches. There are sixteen customary cups per gallon, making a a customary cup 14 ⁷⁄₁₆ cubic inches. Cups of this size can be found in any US supermarket for use in following cooking directions.
Can a conceptual cup as unit of measurement be used to measure the volume or weight of a big solid object that would never fit in a cup?
Yes, theoretically. But when measuring cheese by the cup, you would measure it in the form called for by the recipe, such as cubed, shredded, finely shredded, or grated. So you are actually measuring not the volume of the cheese but the volume of the cheese plus the interstitial air.
What else could cup mean in [cup of cheese]?
In the context of a contemporary US mac-and-cheese box, nothing else. In other contexts, there are many possibilities. For example:
Cup Type Volume (mL) Comments
-------------- ----------- --------------------------------------
US legal 240 nutrition labels
US customary 237
US coffee 118–177 not standardized even on coffeemakers,
but generally 4–6 customary fl. oz.
Turkish water 200–250 “su bardağı”, water glass/tumbler
Turkish tea 100–125 “çay bardağı”, tea glass
Turkish coffee 75–90 “kahve fincanı”, coffee cup
Japanese gō 180
32A 241 brassiere measurement, US system
In older recipes from many countries, you will find volume given in common household units such as coffee cups, teacups, and tumblers. Sizes vary. For example, a cup in a Turkish recipe is not equivalent to a US cup. The same is true for teaspoon and for other measuring devices. These words refer to different size objects according to locale and even time period.
Wikipedia, cooking sites found with Google, and personal experience.