Why a poor old hog get's associated with shaving is a bit odd. Some brushes are labelled lather hogs because lather tends to become embedded in the brush and is a bit reluctant to release it onto your face. It's usually a characteristic associated with dense brushes.
I think a lot of people tend to mash down too heavily in creating lather. It's really best to use the tips of the brush with minimal downward pressure when making lather in a bowl. The right balance between product and water will produce plenty of lather. And when lathering directly on your face the same technique of using the tips and light pressure can be very effective. So if you think you have a brush that's a lather hog, try using a little less pressure. It's worth a try.