How is the belief that Christians are both adopted and begotten sons of God reconciled? If one is begotten, then one cannot be adopted by those same parents. The idea of being begotten means that you are the biological progeny of your parents. In other words, the woman whom you call “mother” is the same woman who gave birth to you. But, in the case of adoption, the woman whom you call “mother” is not the same woman who gave birth to you. (Let’s recall that the New Testament was written in the 1st century A.D. There were no such things as in-vitro fertilization or surrogate motherhood.) Likewise, if you are adopted, your father is not the same man who begat you. With that being said, how can we reconcile the fact that Christians are said to be both begotten/born again and adopted?
Begotten

John 1:12-13 cp. 1 Pet. 1:3


12 But to those who received him, he gave them power to become the sons of God, even to those who believe that believe in his name, 13 who were begotten, neither of blood, nor of the will of the flesh, nor of the will of man, but of God.
ΙΒʹ ὅσοι δὲ ἔλαβον αὐτόν ἔδωκεν αὐτοῖς ἐξουσίαν τέκνα θεοῦ γενέσθαι τοῖς πιστεύουσιν εἰς τὸ ὄνομα αὐτοῦ ΙΓʹ οἳ οὐκ ἐξ αἱμάτων οὐδὲ ἐκ θελήματος σαρκὸς οὐδὲ ἐκ θελήματος ἀνδρὸς ἀλλ᾽ ἐκ θεοῦ ἐγεννήθησαν TR, 1550

Adopted

Gal. 4:5 cp. Rom. 8:15; Eph. 1:5


5 in order to redeem those who were under the Law, so that we may receive the adoption of sons.
Εʹ ἵνα τοὺς ὑπὸ νόμον ἐξαγοράσῃ ἵνα τὴν υἱοθεσίαν ἀπολάβωμεν TR, 1550

Michael Peppard wrote,

In the move from proclamation to narration, however, each early Christian theologian had to make a decision about how to portray the metaphor of divine sonship. Is it begotten or adopted? By the fourth century, the answer would become mostly clear: the divine sonship of Jesus Christ was by begetting and that of everyone else was by adoption. But in the first two centuries, and especially in areas more influenced by Roman culture, a high view of adopted sons was well established.


References
Peppard, Michael. “Adopted and Begotten Sons of God: Paul and John on Divine Sonship.” Catholic Bible Quarterly, vol. 73, no. 1, 2011, pp. 92–110.
In the verse you quoted, John 1:12-13, the Greek words for "children of God" is "tekna Theou." (Notice that "tekna" is always referring to "little children.")

Contrast this with John 3:16 which uses "monogene huion" or "only Son." The term "monogene" being the one containing the notion of "the only offspring" or "the only begotten one." 

(Also, anywhere else in the New Testament, "huion" always refers to a "mature" children, not little ones)

Based on this, I believe that the translation that use the word "begotten" should not be used to ascribe the concept of "begotten" into these verses. In fact in ESV translation (and some others), there is no "begotten" in both John 1:12-13 and John 3:16.

I also believe that the only "begotten" Son is the Lord Jesus Christ and Him alone. Believers are adopted sons (and daughters) of God.  There is no notion of "begotten" ever applied to believers.