The command in Genesis and the elaborate rites of the Grihya Sutras have almost nothing in common except a conviction that the opening moments of a life are not neutral, and that something should be said into them.
In many Muslim households the call to prayer is spoken quietly into the newborn's right ear and its shorter companion into the left, so that the first intelligible sound the child receives is the name of God. The practice comes from the hadith rather than the Qur'an and is treated as recommended rather than required, which has not stopped it becoming near-universal. Alongside it goes tahnik, a smear of softened date on the palate, reported of the Prophet and now often done with any sweet thing to hand.
Vedic domestic ritual is more elaborate and much more explicit about what it is doing. Jatakarma is performed before the umbilical cord is cut. Honey and clarified butter are given, a verse asking for intelligence is breathed into the ear, and the father addresses the child directly. The rite treats birth as incomplete until it has been spoken over.
Christian practice is thinner and later. The Orthodox churches have prayers for the mother on the day of the birth and a presentation of the child on the fortieth day, which follows the Temple presentation in Luke rather than any theory about newborns. Western churches largely folded the whole area into baptism.
What is worth noticing is where the rites are performed and by whom. Almost all of them are domestic, and a great many were carried out by women in the house with no clergy present at all. That is one reason the surviving textual record of birth ritual is so much thinner than the record of anything performed in a temple, and why reconstructions of it lean disproportionately on the few traditions that wrote their household ceremonies down.