Mother told us that too — it's an old wives tale. The truth is, shaving or not shaving, more body hair grows over the same areas as humans of all genders mature physically — hence, thicker growth. Observationally, as for growing darker, Mother and my sister, who were blonde in their youth, had thicker growth and darker hair the older they got, as is the nature of most hair colors. Myself and my two redheaded sisters, on the other hand, did not; all our growth remained red-haired and fairly sparse.
In fact, when I stopped shaving, after a few years, my body hair seems to changed to growing naturally softer and sparser.