God as a human / a nation's father
In his answer, Nigel already covered the Biblical basis for the fatherhood of each human being:
- bodily & existentially ("out of the dust from the ground", Gen 2:7a) and
- spiritually (receiving "breath of life", Gen 2:7b) as being stamped with God's image (which includes in it our conscience and the capacity to know and love God).
This is the common heritage of all humanity: God being every human being's "father".
In addition, the Jews (as biological descendants of Abraham) correctly understood God as the father of Israel who made a covenant with them (Hosea 1:1)
In both verses you cited, the prophet Malachi and the Jesus (in his office as prophet), confronted the Jews that they have not acted faithfully in their covenant with God:
- profaning the Temple by sacrificing to foreign God (Mal 1-2)
- by being driven to sinful acts due to following their sinful desires, shown by their refusal to believe Jesus's words and by their trying to kill God's prophet (John 8:34-47)
Before Jesus came, the Jews understood this God as the "Most High God" (Gen 14:20) above all other gods (monolatry) which over the OT history developed into the belief that this Most High God has ultimate power over other gods, forces of nature, the realm of the dead (the Pharisees believed the resurrection of the body), and even time & space & psyche that no man/woman can hide from (Ps 139).
THE ABOVE IS THE JEW'S ANSWER to your question "who is this father they speak of*.
Trinitarian formula as deeper insight into Israel's God
After Jesus came, we have deeper insight to Israel's God who revealed himself in Jesus, which the apostles and the NT writers understood from his words, his teachings, and his deeds as showing Jesus not only as prophet (Deut 18:15), as king (Eze 37:24-27, Zech 9:9), as messiah (Isa 9:6-7), and as Son of Man (Daniel 7:13-14), but also as the incarnation of Israel's God into 100% human who previously was already somewhat present in the Holy of Holies, though less fully than in Jesus. In Jesus God declares that we can obtain the forgiveness of sin, peace, guidance & wisdom, spiritual water & bread (to nourish our spirit), and especially power to overcome sin, if we repent and identify ourselves with Jesus's death and resurrection in baptism (i.e. born again, John 3:5-8).
Trinitarians believe that Jesus can give us all the above because he IS "the human being of God" who truly walked as God along the shores of Galilee to demonstrate that God is truly WITH US in solidarity in human weakness and frailty. Jesus is Israel's God in bodily form, from whom all grace must come, channeled through Jesus to us as the vine to the branches (John 15:4-5). If Jesus were mere human, it would be preposterous. If Jesus were mere super creature greater than angel, he would still owe his existence to this supreme God, which would risk creating idolatry just like Israels were tempted to worship foreign gods they perceived to be greater than YHWH.
In the NT, Jesus is described as the "Son of God" in a sense MORE than human beings are "sons of God" and MORE than David as "Son of God" (as King). Since Christians believe that the NT authors were inspired when they wrote as they processed their encounter with Jesus (either in person or through the testimony of others who did), Trinitarians then tried to find a good formula / concept to distill in proposition form what NT writers wrote as letters, narratives, etc. Trinitarians do NOT add to the revelation, but through the formulas they used proposition form to indicate truths of the nature of Israel's God as well as to carve out a boundary to exclude errors regarding the nature of Jesus and of God.
Thus, through the early centuries, Trinitarians laboriously came up with a formula preserved in the creeds which are respectful, continuous, and downstream (in terms of prophetic content) from the Bible, that as "Son of God" Jesus is the enfleshment (the adding-on of the bodily human nature) of the "eternally generated" Word from the Father (the Word being spiritual, the second person of the Trinity). It is very important to understand that
- this Father-Son relationship implied by this "eternal generation" is NOT exactly the same as human father to human son relationship. It is an imperfect analogy to describe the immanent relationship within the Godhead. This is done to preserve the divinity of Jesus, so that the practical benefit for us described above makes sense (all grace must come from God directly, only God can transform us to be like Him). It is also to assert that the human Jesus is ONE IN BEING with Israel's God, thus preserving monotheism when we worship Jesus.
- Jesus as human being has another Father-son relationship (in his human nature) with Israel's God which (unlike the eternal generation above) we can fully imitate by grace as fellow human beings, when Jesus prayed to God and depended on God, for example. In this regard, Jesus can truly become our model human being. He is a servant leader (washing the disciples' feet). He forgives enemies with mercy. He obeys secular government when they rule consistent with God's laws. He lays down his life for his friends according to his mission. He is a friend to the downtrodden and social outcast. He is guided and empowered by the Holy Spirit. He obeys his conscience without falling into temptation. Etc.
THE ABOVE SHOWS FURTHER INSIGHT OF ISRAEL'S GOD INNER NATURE understood from the Trinitarian's perspective when the early church fathers processed further the NT records of the Apostles's encounter with Jesus. The resulting understanding IMPROVES ON the Jew's understanding of Israel's God by seeing Jesus not as a separate being, but as the SAME being conceived as God's Son (eternally generated) and as Second Adam (model human being into which we are being transformed by the power of the Trinitarian life infused to us when we are born again).
Since the Jews did not have this improved / deeper insight into Israel's God (because they understood Jesus as a separate being), their understanding of Israel's God as Father is not Trinitarian. Trinitarians "recast" (to use a programmer's term) the Jewish's understanding of Israel's God into 3 "subsistent relations" to whom we (as created beings) relate personally to this One Most High God in 3 hypostases:
- the Father (origination side of the eternal-generation relation) who is the subject prominently acting in the OT
- the Son (filiation side of the eternal-generation relation), who incarnated as Jesus in the NT but also made "cameo" appearances in the OT (as Lady Wisdom in Proverbs 8, for example)
- the Spirit (the spirated-love relation), who also made "cameo" appearances in the OT but more prominently now (since Pentecost) in the life of the church and inside each believer
P.S. (answers to objections)
Objection 1: All this language of hypostases, generation, and spiration reduces God into a concept
Answer 1: Theology uses concepts to describe God as best we can while asserting that we can never define God. Just as OT uses anthropomorphic language to describe a lot of God's actions and attitudes to us, Christian theologians use philosophical language to try to understand the inner nature of God as revealed more fully in the NT. Both are human analogies that we know to be inadequate to fully define God, but we use them to indicate truth and exclude errors so we can relate to the Biblical God better and discard other religions's concepts of God.
God is what He is, God does what He wants. He is ultimately mysterious but not unknowable or incomprehensible. Using concepts we can approach God in our thinking, although we cannot stop at thinking, but the thinking needs to result in AND TO ACCOMPANY (!!) acts of worship, acts of love, repentance, etc (since we are rational beings). We are not mere animals who do without thinking. Thus, we use concepts to understand and in turn, to love with understanding. Even the state of rapture / mysticism / beatific vision is described as beyond thinking, not against thinking. It is very hard to love someone we don't understand. Recounting actions and perceived emotions can only go so far; understanding helps us appreciate what animates the other person: their intention, motivation, and inner desires. Since God is spirit, Trinitarians use the psychological (spiritual) analogies of human knowing and loving to help us understand the immanent life of God as both are quintessential to the human spirit. The theory of subsistent relations, for example, enable us to appreciate the "inner heart" of God to understand his deeds better, as described in the Bible. Thus, philosophical language used reverently can complement the revealed anthropomorphic description of God in the Bible in a way that is consistent with Biblical discourse about God.
Furthermore, our mind uses concepts to direct our reason, will, and emotion toward our creator in a posture of worship, prayer, and thanksgiving and secondly, to motivate us to love ourselves and our neighbors better, as we cooperate with the grace given to us. In theology God is described analogously as faithful, loving, merciful, just, fatherly, etc. Those qualities are more concrete when seen through the lens of subsistent relations since at the core we are relational beings too. During the sanctification process our mind can better reflect and measure our own progress compared to how Jesus has those qualities in their fullness in his human nature as Jesus relate to the Father and as Jesus responds to the Holy Spirit. We want Jesus's Trinitarian life in our hearts as well, and Christianity teaches that God desires to give that life to us! It is our responsibility to receive, to understand (using concepts), and to let the indwelling Trinitarian life to be more influential in our earthly lifespan.