This distinction between sex and gender is a common use of those two terms (at least in particular contexts) but the idea that "man" and "woman" identify genders and "male" and "female" identify sexes is much less common. More commonly, "male" and "female" are adjectives pertaining either to sex or gender and "man" and "woman" are mind referring to adults who are, respectively, male and female, again, either by sex or gender.