I think the answer to your question turns on the underlying assumptions on what it means to adopt someone or for them to be adopted. Are you asking:
- For a legal standing of Jesus adoption by Joseph, wherein a court would accept that Jesus is legally Joseph's son?
- For a simple proof that Jesus was accepted as Joseph's son by his family and by society in general?
I believe we may not be able to adequately establish the first, but the second can be, and I will attempt to establish it in the remainder of this answer.
First, let's establish the simple facts of Jesus birth, from scripture:
18 Now the birth of Jesus Christ happened this way. While his mother Mary was engaged to Joseph, but before they came together, she was found to be pregnant through the Holy Spirit. 19 Because Joseph, her husband to be, was a righteous man, and because he did not want to disgrace her, he intended to divorce her privately. 20 When he had contemplated this, an angel of the Lord appeared to him in a dream and said, “Joseph, son of David, do not be afraid to take Mary as your wife, because the child conceived in her is from the Holy Spirit. 21 She will give birth to a son and you will name him Jesus, because he will save his people from their sins.” 22 This all happened so that what was spoken by the Lord through the prophet would be fulfilled: 23 “Look! The virgin will conceive and bear a son, and they will call him Emmanuel,” which means “God with us.” 24 When Joseph awoke from sleep he did what the angel of the Lord told him. He took his wife, 25 but did not have marital relations with her until she gave birth to a son, whom he named Jesus. (Matthew 1:18-25 - NET)
From the account, we can see that Jesus was conceived within Mary by the Holy Ghost. Joseph took Mary legally as his wife prior to Jesus' birth, though they did not consummate the marriage until after Jesus was born. Based on this, it seems clear that there may not have needed to be any real "adoption" per se, but that Jesus would have been accepted as Joseph's son without any need for adoption. It seems pretty clear that by following the divine directive and marrying Mary that Joseph accepted his place as Jesus earthly father and "adopted" Jesus in a very real way. Only Mary and Joseph would have known otherwise.
From Luke, we know that Mary (and Joseph) are originally from Nazareth:
26 In the sixth month of Elizabeth’s pregnancy, the angel Gabriel was sent by God to a town of Galilee called Nazareth, 27 to a virgin engaged to a man whose name was Joseph, a descendant of David, and the virgin’s name was Mary. 28 The angel came to her and said, “Greetings, favored one, the Lord is with you!” (Luke 1:26-28 - NET)
So Mary was living in Nazareth originally. However, Jesus was born in Bethlehem:
1 Now in those days a decree went out from Caesar Augustus to register all the empire for taxes. 2 This was the first registration, taken when Quirinius was governor of Syria. 3 Everyone went to his own town to be registered. 4 So Joseph also went up from the town of Nazareth in Galilee to Judea, to the city of David called Bethlehem, because he was of the house and family line of David. 5 He went to be registered with Mary, who was promised in marriage to him, and who was expecting a child. 6 While they were there, the time came for her to deliver her child. 7 And she gave birth to her firstborn son and wrapped him in strips of cloth and laid him in a manger, because there was no place for them in the inn. (Luke 2:1-7 - NET)
After this they were apparently in Bethlehem for quite some time, rather than returning to Nazareth. They waited for the wise men to arrive, then fled to Egypt, and eventually returned, after Herod died. Rather than return to Bethlehem, he went back to Nazareth:
19 After Herod had died, an angel of the Lord appeared in a dream to Joseph in Egypt 20 saying, “Get up, take the child and his mother, and go to the land of Israel, for those who were seeking the child’s life are dead.” 21 So he got up and took the child and his mother and returned to the land of Israel. 22 But when he heard that Archelaus was reigning over Judea in place of his father Herod, he was afraid to go there. After being warned in a dream, he went to the regions of Galilee. 23 He came to a town called Nazareth and lived there. (Matthew 2:19-23)
So, it's not entirely clear, but certainly plausible, that those living in Nazareth didn't have any idea about the exact circumstances of Jesus' birth, as Joseph and Mary were not around Nazareth for a substantial period of time. They certainly did not know that Joseph wasn't really the father.
It's clear from later accounts in scripture that the townspeople believed Joseph was Jesus' natural father:
54 Then he came to his hometown and began to teach the people in their synagogue. They were astonished and said, “Where did this man get such wisdom and miraculous powers? 55 Isn’t this the carpenter’s son? Isn’t his mother named Mary? And aren’t his brothers James, Joseph, Simon, and Judas? 56 And aren’t all his sisters here with us? Where did he get all this?” 57 And so they took offense at him. But Jesus said to them, “A prophet is not without honor except in his hometown and in his own house.” 58 And he did not do many miracles there because of their unbelief. (Matthew 13:54-58)
So regardless of what we consider the "legal adoptive status" of Jesus with respect to Joseph, the townspeople, at the very least, accepted him as a legitimate member of Mary and Joseph's family, calling their other children Jesus' brothers and sisters. This seems to corroborate that Joseph's action of marrying Mary was also an act of adopting Jesus from a practical perspective.
I don't know that this can allow us to "legally" call Joseph as Jesus' "adoptive" father, and I am not familiar with Jewish custom or law on this point.