That is like saying my son is me by default
Not by default. Like begets like. Each after it's own kind. It's a law of nature. With man and animals however our children, while similar, are not perfect reproductions. With God however, everything He does is perfect. The Son...His Son... Is a perfect reproduction of Himself, the Son having the exact same attributes as the Father... Divine attributes. The
only difference being that everything Jesus is He received from the Father. This there is only one difference between the Father and the Son... Rank.
Here are some quotes from biblical scholars who can explain the above with far greater erudition than I.
According to this, (quoting John 1:14,18 and John 3:16) Jesus Christ is begotten of God in a sense that no other being is; else he could not be his only begotten Son. Angels are called sons of God, and so also are righteous men; but Christ is his Son in a higher sense, in a closer relation, than either of these. God made men and angels out of materials already created. He is the author of their existence, their Creator, hence their Father. But Jesus Christ was begotten of the Father's own substance. He was not created out of material as the angels and other creatures were. He is truly and emphatically the "Son of God," the same as I am the son of my father.”
The same gentleman concluded by quoting Hebrews 1:1-8, and commented...
By this we see that a very plain and great distinction is made between the Son and all the angels. They are all commanded to worship him. No created being can ever be worthy of worship, however high he may be, neither would it be right nor just for God to bid one order of his creatures to worship another. Divinity alone is worthy of worship, and to worship anything else would be idolatry.”
Here's another...
The Word was “in the beginning”. The mind of man cannot grasp the ages that are spanned in this phrase. It is not given to men to know when or how the Son was begotten; but we know that He was the Divine Word, not simply before He came to this earth to die, but even before the world was created.... We know that Christ “proceeded forth and come from God” (John 8:42) but it was so far back in the ages of eternity as to be far beyond the grasp of the mind of man”
“The Scriptures declare that Christ is "the only begotten son of God." He is begotten, not created. As to when He was begotten, it is not for us to inquire, nor could our minds grasp it if we were told. The prophet Micah tells us all that we can know about it in these words, "But thou, Bethlehem Ephratah, though thou be little among the thousands of Judah, yet out of thee shall He come forth unto Me that is to be ruler in Israel; whose goings forth have been from of old, from the days of eternity." Micah 5:2, margin. There was a time when Christ proceeded forth and came from God, from the bosom of the Father (John 8:42; 1:18), but that time was so far back in the days of eternity that to finite comprehension it is practically without beginning. But the point is that Christ is a begotten Son and not a created subject.”
Either Christ is truly the Son of the living God as Peter declared, or He isn't.