When I was a vegetarian, back in the day, I learned that the enzyme that makes milk turn into cheese is rennet. the rennet enzyme comes from aminal stomachs. So strict vegetarians would only eat rennetless cheese, which is made from a new rennet enzyme developed from plants.
Of course milk is an animal product. There are vegetarians that do milk and eggs but, generally speaking, a vegan, by definition, would not do milk or eggs. Maybe they really are milking the nuts?
Art has a double face, of expression and illusion.