Genjou Sanzou would deny that he ever thinks about whose child he might have been, and it is true; it is something he consciously denies himself, and that he has trained into himself since the early days at the temple, with bare legs in the cold wind, sweeping leaves off the path.
He decided then that he might as well say that the river is his mother, and Koumyou his father, in every way that matters; that Koumyou plucked him from the earth and water of matter (mother), and raised him to the heavens.
Of course, to say that you "might as well say" something implies that you think about it, and he doesn't think about it.
Though there must have been something odd going on. Parents may cast a daughter to the waters, but there are very few people who would abandon a son. Perhaps she wasn't supposed to be pregnant. He looks at the women round Gojou, looks at Hakkai-the-image-of-his-sister, looks at Kougaiji's alchemist and Kougaiji's sister. He very deliberately doesn't imagine a young girl, still unmarried, who has been binding herself for months in an attempt to hide her swelling waist, who has screamed through childbirth somewhere that nobody could hear her, who goes running down to the river by night with the wet grass brushing her ankles, the basket in her arms a weight that she cannot bear to put down.
It wouldn't matter even if it had been that way. It makes no change to who he is.
It wouldn't matter if it had been an older woman who walked down to the river in the dawn light, her robes trailing in the morning dew. A captive of bandits, perhaps, trying to keep the last relic of her husband alive in this little scrap of newborn flesh and blood, casting her child to the mercy of the river and the gods. Watching the basket float downstream. Turning back again, and throwing her sash over the branch of a tree.
It still makes no change. Why does a Sanzou take a new name? Because he is a Sanzou now, and what he was before that is no longer relevant. Whoever it was cast him onto the waters and let go of him. He lets go of them in return, lets them vanish into the distance, and even thanks them for it, for not being there and not needing something from him.
His mother was the river. It flows behind him, washing his past away.