Can a human coherently claim to be "God the father"?

Pretty much all Christians (Trinitarian and not) would agree that the Father has not and will not become incarnate. From now on I'll just speak for Trinitarians: the incarnation was a singular event for the purposes of:

1. The full self-revelation of God (Hebrews 1:1-4)
2. the mediation between and reconciliation of God and humanity
3. the salvation of humanity from sin
4. and the uniting of humanity to God (the doctrine called Union with Christ)

These are the crowning achievements of the Son. He has accomplished them, and so it would be both unnecessary and kind of dishonouring to the Son for the Father to also become incarnate. We are united to God and we relate to God through Jesus Christ, the Son of God, our king. This is a singular kind of role, often called the threefold office of Chirst, for a singular incarnate God-man.

In the land of hypotheticals however, there is disagreement whether it was purely arbitrary which person of the Trinity would become incarnate. If it was arbitrary then the Father could have become incarnate instead of the Son. But if it was not arbitrary and there is something ontologically fitting for the Son to become incarnate that does not apply to the other persons of the Trinity, then in all hypothetical universes it would only be the Son who became incarnate.

I'm in the second camp: I think that the way the scriptures describe the Son of God as the Word of God has the connotation that, although all three persons are involved in all the works of God, it is primarily the Word of God that accomplishes his works. Think of creation being spoken into existence and life in Genesis 1, or how God chooses to relate to his people primarily through covenants, speech acts which redefine the reality of God's relationship with his people through the very act of being spoken, creating a people and giving them the life of God. The incarnation is the physical personification of the Word of God in creative life-giving action. So for this reason I think it is particularly fitting that the Word (Son) became incarnate as he is the one who has the front role in accomplishing the works of God.

The starting point for my answer is in the Old Testament prophecy of Isaiah where there is an intriguing link between a son to be born (who will have the title 'everlasting Father') and God already having the same title. The setting is the prophecy of a great light to be seen by Israelites walking in darkness.

"For unto us a child is born, unto us a son is given: and the government shall be upon his shoulder: and his name shall be called Wonderful, Counsellor, The mighty God, The everlasting Father, The Prince of Peace.' Isaiah 9:6 A.V.

"The remnant shall return, even the remnant of Jacob, unto the mighty God." & "Thou, O Lord, art our father, our redeemer; thy name is from everlasting." Isaiah 10:21 & 63:16 A.V.

The Trinity doctrine finds no difficulty with this, given that the foretold son would be the unique, only-begotten Son of God, who existed in the bosom of God the Father from eternity:

"No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him." John 1:18.

This rules out the idea of some who think the man, Jesus, had no existence prior to being born of the virgin Mary. This accords with the New Testament revelation of the Word of God, who made everything that was made, incarnated as the man, Jesus - fully God and fully man. That is why he was called both the Son of Man and the Son of God.

However, God the Father remained in heaven while the Son incarnated here on Earth. This is shown in the New Testament when, at Jesus' baptism, the Holy Spirit appeared over his head as a dove, and the Father's voice was heard from heaven saying, "This is my beloved Son, in whom I am well pleased." This is why, a few verses further on, the link is made between Isaiah 9:1-2 foretelling this son, and the Isrealites sitting in darkness seeing a great light - for the Light of the world had appeared to them, as Jesus Christ. See Matthew 1:20-23; 3:17; 4:13-16 and Luke 1:31-33 & 76-79; 2:30-32.

This means that it was not any mortal man who took the title of 'God the father' (which would be blasphemy), but the eternal Son of God who came to Earth as God incarnate, adding human nature to his divine nature. That, however, in no way constituted any transfer of God as Father to the Son of God. The Father God remained in heaven while the Son incarnated, with the Father's approval. The Christ (as Word of God) was equally involved in creating everything and in being our redeemer (Romans 8:14-21, Ephesians 1:10-14 & Luke 24:21).

The Trinity doctrine shows three Persons sharing the one divine nature. The Father and the Son share the one, divine nature, with absolute unity of the Spirit in that nature. What God is in nature, the Son and the Holy Spirit are, too, without any dilution, or transfer of aspects of that nature, one to another. They equally share all of that divine nature, the revelation of which is opened up to us in the New Testament, and by the Spirit's revealing.

So, it would be blasphemy for any mere human to claim the title, "God the Father", but the scriptures show the Son of God to equally be the source (Hebrew ab = father) of our human and spiritual lives as is Father God.