Birth of a child (labour) is роды. Always plural. Принимать роды = act as a midwife, obstetrician. Hence, рожать = give birth to.
That is the table for Роды роды — Викисловарь


Род means something of the following: gender, class, type, tribe or clan (so род is polysemantic). If it's tribe or clan - all born to that tribe or clan (весь род - all people born to the clan).

You see that the root is the same "род" - the etimology is the same.