Some old hymns are a poor source of obtaining one's theology from. New hymns are even worse. Modern choruses and 'praise song' can be downright atrocious. I have many worship music books, with words, and I have penciled through lots of wrong words and phrases, writing in more theologically correct ones. Then I can sing them as I play piano. Unfortunately, I often go to worship services and have to simply not sing some of the lines, or verses. Not so with the Psalms of course, but that's an even more volatile matter in some circles!
The two scriptures quoted in your question form part of the answer. Christians are called children of God in the New Testament. Jesus taught us to pray to our Father in heaven. Yet Messiah is also the "Everlasting Father"! We are may things to Jesus, and he is many things to us. We live in him, and at the point of faith when that starts to happen, he can be viewed as giving us 'new life'; we are born again of the Spirit; then we can call God "Abba, Father" (Romans chapter 8). A whole raft of metaphorical language is employed in the scriptures to give us some idea of the vastness of our new relationship to God, in Christ, through the Holy Spirit.
However, an important point is that Jesus Christ is called the only-begotten Son of God, and never "the only son". Some modern translations are misleading here, for God has more than one son! However, he only has one only-begotten Son, who was never created. That's the point. All other sons of God had a starting point in time, being created by God (and don't overlook how God the Word made everything that was made (John 1:3). This means that God the Father and God the Son (the 'Everlasting Father' of Isaiah 9:6) share that way of relating to those who are called "sons" in the New Testament.
The question deals with how we relate to Christ, and the biblical use of calling Christ 'the only begotten Son, which is in the bosom of the Father, he hath declared [God]' (John 1:18) shows how Christ relates to the Father. Theirs is an utterly unique relationship that has no starting point in time. We, however, need a starting point in time, both to begin existing physically, and to start being 'children of God', born of the Spirit. John's gospel goes into this. For instance, John wrote,
"But as many as received [Christ], to them gave he power to become the
sons of God, even to them that believe on [Christ's] name, which were
born, not of blood, nor of the will of the flesh, nor of the will of
man, but of God." John 1:12-13 KJV
That is what brings us into a spiritual relationship with God and Christ. Then we relate to them as in a family - we are brought into the family of God, and that should open us up to understanding the astounding and unique relationship between the eternal Father and the eternal Son, and how we then relate, as children, in that spiritual family.
To ask, Are we children or brethren of Christ? is to fall short of asking the question that would give the answer. It's not "either", "or" - it's both. There is no contradiction here because this is all about a spiritual relationship that goes beyond any relationships we have as mortals on earth. It is vastly more complex and wonderful and priviledged than any human relationships, though they are often used in an illustrative way to help us start to understand.
