The term "father" identifies sufficiently the role of God to the believers. "Father" carries with it a level of dignity and respect that the term "daddy" does not. "Daddy" is a term of familiarity at a level that is not shown in the scriptures of one's walk with God. We are to be child like in some respects, but not childish.
Daddy may seem childish to you, but that is not how everyone views it. I'm not quite sure how to explain this, but when I use the term daddy it is like I'm making myself vulnerable. Dad is a title given to my father in place of his name, because I was raised to believe children do not call their parents by their names.
Daddy is a term of endearment. It isn't a title. Like a wife calls her husband "honey," I call my dad "daddy." It encompasses the love and dependance I have on my father. It tells him "no matter how old I get, you will always be my daddy. You will always be my pillar I can turn to for security, because I know that when I need someone to shelter me, you are there."
It's the same for God, except more complete. My earthly daddy is not perfect, and he cannot shelter me from everything. God can. When my world seems to be crazy, I know my Daddy God is there to hold me and comfort me and fend off anyone who tries to harm His princess.
So, you can call it childish, but don't dare to assume that something as personal as the term "daddy" is childish to me. It is so much more than that. So much more than you could ever possibly understand.