There's really no one religion that only eats the placenta.
Many religions eat the Placenta after birth if they believe in it.
The religion beliefs behind the practices of eating the placenta, whether in part or in whole, commonly reflect acknowledgment for the vast work of this organ for the baby in utero, serving as its 'protector' and providing critical vital functions for the baby before birth.
The human placenta can be seen as the Tree of Life, as a genetic 'twin' to the fetus, an angel, and reasons for ingesting the placenta may reflect spiritual beliefs as much as the pragmatic ones listed above.
The traditional practices to revere and honor the placenta that do not include consumption may include placenta burial, such as in Saudi Arabia.
Such traditions reflect human birthing practices wherein umbilical cords may not have been severed while the cord is still pulsing, avoiding blood loss and infection, and may include practices that retain the placental connection until after it has been delivered and the baby is already nursing.
The Chinese traditional medicine has for centuries used human placenta to treat kidney and liver ailments or low energy, though not in postpartum mothers.
Eating the placenta after birth can cause health issues for both the parent and the baby so it's not really good to eat the placenta.
I thought it was gross to eat the placenta and to me it still is.