Plants generally capture water through their roots which typically spread evenly in all directions, water travels down through the soil until it hits a ledge then travels along its path (called percolation.) That means a rough measure of the amount of water is plenty to guage requirements for shallow rooted plants, like most food crops. An acre is a measure of area and a foot is a measure of length, combined those give the 3 dimensions to make volume. Water is sold by volume, and for agricultural purposes that is the measure it is sold by to farms (typically.)

Acre-feet is an agricultural term so that's probably why you haven't heard of it. Feet of water is generally used to describe how much water a plant needs, acre-feet are how much water a crop needs (almonds need 4 acre-feet of water annually in California.)