Let’s not overthink this. Without any other information given, the “cup" referred to is assumed to be a standard measuring cup in the American style measuring cup type used in cooking, which contains, when full to the “one cup” measure line, 8 ounces of a substance, whether liquid or dry substance. A cup is a cup, and a cup is 8 ounces. The measuring vessel must be assumed to fit the substance being measured, appropriate to its type.
Whether a dry measure, such as metal or plastic marked full by filling completely and leveling the substance at the top edge by scraping excess off with a straight edge such as the back edge of a common table knife which has such an edge; or a liquid measure, which would be clear glass or clear plastic cup marked off on the side in red markings, which can only be accurately filled and read off clearly by pouring liquid inside it until reaching the desired marking; each is one cup when full to that mark.
Especially when the questioner is looking for the amount of “1/4 of a cup in ounces.” Obviously it’s not weighted, as that has no relation to the style, size, or type of container or even the type of substance inside.
One cup is 8 ounces. 1/4 of 8 is 2.