Well, the sacred writer, St. John, removed any future ambiguity in the reading of such passages already, in chapter 5:
John 5:18 (DRB)
Hereupon therefore the Jews sought the more to kill him, because he did not only break the sabbath, but also said God was his Father, making himself equal to God.
Or simpler yet:
In the beginning was the Word, and the Word was with God, and the Word was God.
How can Jesus, "the Word . . . made flesh," be both "with God" and "[be] God"? Jn 1:1,14 Because they are distinct Persons in God.
Notice this (5:18) is St. John's inspired 'commentary' on the motives of the Jews. Separate from the narrative, but an explanation of it. Jesus didn't 'allegedly, according to the Jews' break the Sabbath (no less, in the context of saying He, just as God the Father, "works even [on the Sabbath]" Jn 5:1—a claim to divinity itself). He didn't 'allegedly, according to the Jews' "[call] God His own Father, making Himself equal with God." He did.
Why does Father and Son mean more than an 'adoptive' sonship and 'adoptive fatherhood?' (i.e. why does Son mean of the same divine nature of God) Because Jesus claimed to be the Son of God because He literally came forth from Him (this is why He is called the 'Son' or even 'Word' of God in the first place):
"Jesus therefore said to them: If God were your Father, you would indeed love me. For from God I proceeded, and came; for I came not of myself, but he sent me:"
I came forth from the Father, and am come into the world: again I leave the world, and I go to the Father. His disciples say to him: Behold, now thou speakest plainly, and speakest no proverb.
Notice Jesus is not speaking figuratively, but of His actual nature as a Person.
This echoes the words of the eternal Wisdom of God from Sirach 24:5 (vul):
Wisdom shall praise her own self, and shall be honoured in God, and shall glory in the midst of her people, 2 And shall open her mouth in the churches of the most High, and shall glorify herself in the sight of his power, 3 And in the midst of her own people she shall be exalted, and shall be admired in the holy assembly. 4 And in the multitude of the elect she shall have praise, and among the blessed she shall be blessed, saying: 5 I came out of the mouth of the most High, the firstborn before all creatures: 6 I made that in the heavens there should rise light that never faileth, and as a cloud I covered all the earth: 7 I dwelt in the highest places, and my throne is in a pillar of a cloud. 8 I alone have compassed the circuit of heaven, and have penetrated into the bottom of the deep, and have walked in the waves of the sea, 9 And have stood in all the earth: and in every people, 10 And in every nation I have had the chief rule: 11 And by my power I have trodden under my feet the hearts of all the high and low: and in all these I sought rest, and I shall abide in the inheritance of the Lord. 12 Then the creator of all things commanded, and said to me: and he that made me, rested in my tabernacle, 13 And he said to me: Let thy dwelling be in Jacob, and thy inheritance in Israel, and take root in my elect.
14 From the beginning, and before the world, was I created, and unto the world to come I shall not cease to be, and in the holy dwelling place I have ministered before him. 15 And so was I established in Sion, and in the holy city likewise I rested, and my power was in Jerusalem. 16 And I took root in an honourable people, and in the portion of my God his inheritance, and my abode is in the full assembly of saints.
Now clearly, this is the classic divine Wisdom. We might say 'Word,' since He comes from the mouth of the Most High. For Jesus to be using this language of coming forth from God would right be understood to be blasphemous for a mere man to say (but which Jesus never was).
But we also see that for Jews, calling someone the Son of God meant you were calling them divine. They must have understood that for someone to be claiming to be THE Son of God, like Jesus did, He was claiming to be of the same nature of God the Father (hence a 'Son'; hence a 'Father' in the first place).
The Jews answered him: We have a law; and according to the law he ought to die, because he made himself the Son of God.
They realized Jesus wasn't claiming the same adoptive sonship of all of God's people (how could they not, given the way Jesus spoke of Himself, anyway?)
We see Son of God being interpreted as simply 'God Himself' in the following passage also:
The Jews then took up stones to stone him. 32 Jesus answered them: Many good works I have shewed you from my Father; for which of these works do you stone me? 33 The Jews answered him: For a good work we stone thee not, but for blasphemy; and because that thou, being a man, maketh thyself God. 34 Jesus answered them: Is it not written in your law: I said you are gods? 35 If he called them gods, to whom the word of God was spoken, and the scripture cannot be broken; 36 Do you say of him whom the Father hath sanctified and sent into the world: Thou blasphemest, because I said, I am the Son of God?
This is actually a bit of an emphatic linguistic device (God vs. Son of God) by Jesus here: He uses the title 'Son of God' which rings in the ears as something lesser than simply 'God;' so that when He uses it of Himself, it makes them look ridiculous to claim that He can't even call Himself the Son of God, even though He comes forth from the Father and into the world, and is legitimately the eternal Son of God, but they'll allow and accept that mere men were called 'gods' in the Old Testament because of their divine offices as judges. Even though He is the Messiah and the Son of God.
One would be astute to observe that Jesus reacts not as if they had
lied in saying He made Himself out to be God, for which they would be rightly upbraided, and Jesus would have had to have corrected, not played along with. He attempts, rather, to establish the reasoning behind their outrage. And He uses a didactive approach from the Scriptures to show how He has more right to be called God, because He is the Son of the Living God, the Word of God Himself, not one to whom God's word is sent.