Protestants understand that God, as Father, is a revelation of His very nature. God did not choose the persona of a Father – it is who God is. From eternity God has been in relationship with His Son, the Word, whom he begat (not created). They have co-existed throughout eternity, from before creation or the founding of the world:
In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning. Through him all things were made; without him nothing was made that has been made. In him was life, and that life was the light of men. The light shines in the darkness, but the darkness has not understood it.... The Word became flesh and made his dwelling with us. We have seen his glory, the glory of the One and Only, who came from the Father, full of grace and truth” (John 1:1-5, 14).
God the Father sent His one and only Son, to dwell with us. Through the Son we get a glimpse of the eternal relationship between the divine Father and the divine Son. Christ Jesus is God incarnate – Immanuel, a masculine Hebrew name meaning “God with us” or “God is with us.” https://www.gotquestions.org/what-does-Immanuel-mean.html
Through Him we are adopted into God’s family and are privileged to enter into fellowship with God as “Abba, Father”. Adam and Eve enjoyed fellowship with God up until the moment they disobeyed Him and sin entered into the world. But God the Father, through Christ Jesus the Son, has done all that is necessary for that relationship to be restored. This is the significance behind the frequent references in the Bible to God as Father.
Baptist minister Charles Spurgeon touched on this briefly in his sermon no. 2560, delivered at the Metropolitan Tabernacle, London, on 1 February 1883: "I speak that which I have seen with My Father: and you do that which you have seen with your father." John 8:38.
“The relationship of father and son among man implies that one exists before the other, but it is not so implied in the relationship of the eternal Father and Son. We know not how to explain this great mystery, for the terms Father and Son are only the nearest approximation that can be given to our poor understandings of the relationship which exists between them. Yet is the Father eternal and the Son eternal—the Son co-equal and co-eternal with the Father. Our Lord had an existence before He was born of Mary—He had an everlasting existence. His goings forth were from of old, even from eternity. Though He is unto us the Child born and the Son given, yet He is equally, "the Everlasting Father," who was and is and ever shall be One with the eternal God.” Source: https://ccel.org/ccel/spurgeon/sermons44.xi.html
The following extract from the Westminster Confession of Faith gives the Protestant view of God:
Westminster Confession of Faith Chapter 2: 1. There is but one only, living, and true God, who is infinite in being and perfection, a most pure spirit, invisible, without body, parts, or passions; immutable, immense, eternal, incomprehensible, almighty, most wise, most holy, most free, most absolute; working all things according to the counsel of his own immutable and most righteous will, for his own glory; most loving, gracious, merciful, long-suffering, abundant in goodness and truth, forgiving iniquity, transgression, and sin; the rewarder of them that diligently seek him; and withal, most just, and terrible in his judgments, hating all sin, and who will by no means clear the guilty. https://opc.org/wcf.html#Chapter_02
From ‘The J.I. Packer Collection’ by Alister E. McGrath in the chapter ‘On Knowing God’ pp 141-158, Packer  had this to say:
...to Calvin knowledge of God is knowledge within a covenanted relationship. Knowledge of a covenant God who has given himself to you as your God is basic to Calvin's understanding and to the Bible's understanding. Knowledge of God means knowledge of God as the one who has given himself to you. 'Religion', says Luther, 'Is a matter of personal pronouns, I being able to say to God, "My God" and I knowing that God says to me "My child". It is in that relationship that knowledge of God becomes a reality...
The New Testament views knowing your Maker as your Father, and yourself as his child and heir, as the highest privilege and richest relationship of which any human being is capable. Not to know God in this way is, by contrast, to be in a state of fallenness and guilt, cut off from God's life, exposed to his judgment, and under demonic control, whence flows only misery. But this is every man's natural condition.
Jesus said, 'I am the way, and the truth, and the life; no one comes to the Father, but by me (John 14:6). It is as if he said: Yes, a filial relationship to God is possible through relating to me and my mediatorial ministry - though not otherwise. For sonship of God, in the sense that guarantees mercy and glory, is not a fact of natural life, but a gift of supernatural grace. "To all who received him [Jesus], who believed in his name, he gave the power to become children of God" (John 1:12).
The Bible reveals God’s personality and attributes so that we may know Him, in order that we may worship Him, in order that we may have fellowship with Him. From Scripture we can deduce that God is knowable, but incomprehensible. We can know Him but that does not mean that we can understand Him fully or completely. Bearing in mind that God is Spirit (John 4:24), that He is neither male nor female, and that the human mind is too small to span or grasp the essential nature of God’s being, God has chosen to communicate with humans in terms they can understand:
God, in order to tell us about Himself, has spoken a language that you and I can understand. It is almost impossible for us to grasp the idea of infinity and spirituality, so God speaks as if He were a man. He is only doing it that we might understand; so that we may know and trust Him. He speaks as if He were a man – that is the whole idea of ‘anthropomorphism’. And so let us sum up. God is invisible, without parts, without body, free from any and every limitation. Source: ‘God the Father, God the Son’ by Dr. Martyn Lloyd-Jones (1966) – Chapter 5 pages 54-55 
Using human characteristics to describe God is called ‘anthropomorphism’. This is simply a means whereby God can communicate truth about His nature to humans. Since humans are physical, we are limited in our understanding of those things beyond the physical realm. Anthropomorphism in Scripture helps us to understand who and what God is. However, we must not fall into the trap of seeing this as a human attempt to explain, or rationalise, God. When God describes Himself in Holy Scripture, this is not an attempt by human authors to explain a transcendent God. No, it is God who condescends to reach into our finite minds with words and concepts that we can understand. No human can grasp the immensity of the Being of God. Language is God’s chosen means of telling us what He is pleased to reveal of himself to us. And all Scripture is given by the inspiration of God (2 Timothy 3:16) and holy men of God spoke as they were moved by the Holy Spirit (2 Peter 1:21).
Baptist minister Charles Spurgeon delivered a sermon (#2013) on the infallibility of Scripture. “The mouth of the Lord has spoken it” (Isaiah 1:20). All Scripture, being inspired of the Spirit, is spoken by the mouth of God. https://www.spurgeongems.org/sermon/chs2013.pdf What we read about God in the Bible is therefore what God wants us to know about Him.
In the Old Testament God is referred to as a king (Psalm 24:10; 47:2; Isaiah 44:6). Also in the New Testament: “Now to the King eternal, immortal, invisible, the only God, be honour and glory for ever and ever. Amen” (1 Timothy 1:17).
He is sometimes described as a husband (Isaiah 54:5; Hosea 2:2, 16, 19) and throughout the Old Testament his chosen people, Israel, is likened to a wife, oftentimes unfaithful.
In Isaiah 66:13 God is likened to a mother comforting her child, but this is a simile – God is not saying He is a mother. Isaiah 49:15 is another verse that mentions a mother in a description of God, but it is not even a comparison; it is a contrast: God cares more for His people than a nursing mother does her baby.
God as Father can be found in Deuteronomy 32:6, Malachi 2:10 and 1 Corinthians 8:6, to mention just three verses. Jesus taught his followers to pray specifically to God as “our Father” (Luke 11:2) and he himself addressed God as “Abba, Father” (Mark 14:36). How humbling for us that God, in his infinite grace and love, should stoop down to enable us to be adopted into His family:
But when the time had fully come, God sent his Son, born of a woman, born under law, to redeem those under law, that we might receive the full rights of sons. Because you are sons, God sent the Spirit of his Son into our hearts, the Spirit who calls out, “Abba, Father.” So you are no longer a slave, but a son; and since you are a son, God has made you also an heir” (Galatians 4:4-7).
The Spirit himself testifies with our spirit that we are God’s children. Now if we are children, then we are heirs – heirs of God and co-heirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory” (Romans 8:16-17).
Why does God want us to relate to Him as Father? Dr. Martyn Lloyd-Jones sums it up beautifully. Speaking of God, he describes Him as:
...this infinite, absolute, sublime, transcendent, glorious, majestic, mighty, everlasting being who is Spirit, who is truth, who dwells in light no one can approach, this God has been graciously pleased that you and I should know Him, that we should talk to Him and that we should worship Him. Source: ‘God the Father, God the Son’ by Dr. Martyn Lloyd-Jones (1966) – Chapter 5 page 56 
Finally, you ask how God as Father is viewed in general Protestant traditions and if there are significant variances across these traditions. I have neither the time nor the inclination to delve into that aspect, but would simply say that I doubt any Trinitarian Protestant denomination would disagree with the concept of God as Father even though God is Spirit. I am unaware of any orthodox Protestant teaching that views God (or the Holy Spirit) as a “mother figure”. That comes from “new age” spirituality which is not Christian. https://www.gotquestions.org/new-age-movement.html
When Christians pray “Our Father who art in heaven...” it’s because Jesus told us to pray to God that way. As born-again believers we have now entered into relationship with our creator who welcomes us into his family as sons and daughters, heirs to the promise and co-heirs with Christ. This is a spiritual reality for all who are indwelt by the Spirit of God/Christ/Holy Spirit.
 Information on J. I. Packer: https://en.wikipedia.org/wiki/J._I._Packer
 Dr. Lloyd-Jones was a qualified physician who, in 1927, left medical practice and became a Protestant minister in Aberavon (Port Talbot) in South Wales. Then, for thirty years, until he retired in 1968, he ministered at Westminster Chapel in London. He was influential in the Reformed wing of the British evangelical movement in the 20th century.
Note: All quotes that are emphasised with bold/italicised print have been added by me.