Before defining wind velocity, I think it is necessary to explain the continuum assumption. The basic idea is that even though gases are composed of discrete molecules occupying a small fraction of the total volume filled by the gas, gas flows are made up of many individual collisions between gas molecules. In most applications, the flow field is assumed to behave as a continuum and the discrete molecular nature of gases is ignored. By doing so, fluid properties (e.g., density, velocity) can be defined anywhere in the flow.
In order for the continuum assumption to be appropriate, the length scale of the application must be much larger than the mean gas particle path length. In most atmospheric conditions, including in anemometers, the mean path length is on the order of micrometers, while the anemometer size is much larger.
While most anemometers (cup, vane, laser, acoustic resonance) measure wind speed some measure the wind's pressure (e.g., tube, plate anemometers). In the case of cup anemometer, like any other measuring device, there is an error associated with the measurement that depends primarily on the cup size. In general, the anemometer factor (the ratio of wind speed to the rotation speed of the cups) has a value between two and a bit more than three and it depends on the size of the cups and arms.