Yes, almost all mammals eat at least a portion of the placenta following birth, including cows. Preparing a small portion for a human mother to eat following childbirth was a standard practice of midwives historically as it is very effective in reducing or stopping uterine hemorrhage.
It's a good thing to know; this simple act can save a woman's life. Nowadays, a small portion is typically blended with the patient's fruit or vegetable juice of choice to provide a more palatable preparation.