Of course, the bible teacher or pastor's purpose in explaining the word abba this way is to show us that Jesus had a very intimate relationship with his father, not stoic or merely positional. It is what a loving father has with his son and the son who lives securely and comfortably in that love. It is an important message—and it is true.
You can't read John 17, Jesus' intimate and passionate prayer to his Father the evening before his brutal and sacrificial death, and not see this tender intimacy. You see it also in John 1:18 where some versions have it that Jesus dwells "in the bosom of the Father." Ask someone you know well if you can sit at their side. They will be happy and honored to have you do so. Ask them if you can dwell at their bosom and you'll get a different reaction. We also see this Father/Son intimacy at Jesus' baptism where the Father proclaims from heaven to us all his extravagant love and pride in his Son.
This intimacy and love between the divine Father and his Son is as true as the existence of God himself, for it is his very nature. But it is simply not true that Jesus' use of the word abba means something a small child would utter in reference to his father. It does not mean "daddy" or "papa."
This origin of this understanding is generally traced to the notable German Lutheran New Testament scholar Joachim Jeremias who in his 1971 text New Testament Theology explained that abba was "the chatter of a small child. . . . a children's word, used in everyday talk" and seemingly "disrespectful, indeed unthinkable to the sensibilities of Jesus' contemporaries to address God with this familiar word" (p. 67). While Jeremias did not use the word "daddy" or "papa" in relation to abba, the implication was strong and others came along to make that connection.
But other Hebrew and New Testament scholars have taken exception with this understanding.
University of Fribourg's Georg Schelbert critiqued Jeremias' assertion in a 1981 essay and then later in a 2011 book-length treatment entitled ABBA Vater. He contends that Jeremias' interpretation is in "error" and "unwarranted." He elaborates,
In the Aramaic language of the time of Jesus, there was absolutely no other word [than Abba] available if Jesus wished to speak of or address God as father. Naturally such speaking of and addressing thereby would lose its special character, for it is then indeed the only possible form!
This is because, as we shall see, abba means either "father" or one's own father. Schelbert explained that Jeremias even adjusted his earlier understanding in the face of critical peers.
Schelbert was followed by Professor Geza Vermes, a most important scholarly voice on the Jewishness of Jesus. In his book, Jesus and the World of Judaism (Fortress, 1983), Vermes calls out the "improbability and incongruousness of the theory" and that "there seems to be no linguistic support for it." (p.42). Vermez holds, in agreement with Schelbert, that abba can either be understood as "the father" or the more personal, "my father."
This criticism was followed up a few years later with an essay in the Journal of Theological Studies by James Barr (vol. 39, 1988). His article, "Abba Isn't Daddy" explains:
It is fair to say that abba in Jesus' time belonged to a familiar or colloquial register of language, as distinct from more formal and ceremonious language. . . . But in any case it was not a childish expression comparable with 'Daddy': it was a more solemn, responsible, adult address to a Father. (p. 46)
Although he explains that in Jesus' time, this address was used by a father's children of all ages, young and adult, it was often used by small children. Barr adds,
If the New Testament writers had been conscience of the nuance 'Daddy' they could easily have expressed themselves so; but in fact they were well aware that the nuance is not that of 'Daddy' but of 'father'." . . . [T]he semantics of abba itself [based on various evidences] all agree in supporting the nuance 'father' than the nuance 'Daddy'." (p. 38)
It is important and true to understand that God is our intimate Father. So many places in the New Testament make this vividly and encouragingly clear. It is one the rich qualities that makes Christianity distinct from all other faiths and philosophies.
But let's not illustrate this grace for others with something that is not true.
Other articles in this series:
C.S. Lewis and G.K. Chesterton Quotes
Burning Your Ships for Jesus
Misquoting Francis of Assisi
The Cross an Electric Chair?
Divorce Rate Among Christians
Do Faithful Christians Take the Bible Literally?
Is the 'I Only Need Jesus!' Declaration Christian?
Who Really Started the Family 'Culture War'?
Are Your Kids Likely to Lose Their Faith?
Are Millennials More Self-Sacrificing and Community-Minded Than Previous Generations?