Yes, He is the source of the attributes and qualities we know as fatherhood. They're derived from Him, in a logical sense. I prefer to see them as gifts, because they're from the heart.
The wrong-headed way to configure it (I believe we're in agreement) is that man is reading back his fallen understanding of fatherhood on to God ie mankind is the author of the paternal nature.
That way of thinking lands you back with the 'unknown god' (the Pauline springboard at the Areopagus), and worse, denies the core principle of Christian theology: God the Father.
Now suppose one of you fathers is asked by his son for a fish; he will not give him a snake instead of a fish, will he? Or if he is asked for an egg, he will not give him a scorpion, will he? If you then, being evil, know how to give good gifts to your children, how much more will your heavenly Father give the Holy Spirit to those who ask Him?
(Lk 11:11-13)