Used a couple of US recipes recently and most of the ingredients are in cups, or spoons, not by weight. This is a nightmare to convert. Do Americans not own scales or something? What's the reason for measuring everything by volume?
The things people drink out of are many different sizes of course, but when the word "cup" is used in the context of a measure of volume, then yes, they're called "measuring cups", and the volume is standardized.
Same thing with teaspoons and tablespoons. They're not just any random spoon - when talking about measurements, they have a standardized volume and you need to use a cheap and ubiquitous measuring device if you want to follow a recipe precisely.
Most people in USA do not have a scale in their kitchen, but we do have a measuring cup and a set of measuring spoons.
A measuring cup is a specific size, about 237mL. There's a whole system of US measurements, actually:
3 teaspoons in a tablespoon
2 tablespoons in an ounce
8 ounces in a cup
2 cups in a pint
2 pints in a quart
4 quarts in a gallon
Not all cups are measuring cups; if you are having a cup of coffee that doesn't mean your cup is exactly 8oz. You just infer from context that if someone is talking about ingredients then you should measure them with a measuring cup. (Very commonly you also see cups with graduated markings, which are US Imperial on one side and metric on the other, that go up to 2 cups/500mL.)
fluid ounce, since most liquids used in food are nearly the same density.
/edit to add to this, after a cup most things that are dry are not measure in pints, quarts or gallons. For example, you don't hear anyone say "you'll need 1 pint of flour", they'll just say 4 cups.