I interpret this as saying that Jack is still a kid (an innocent) at heart. Pure of heart.
In the short story, both Ennis and Jack are described in 1963 as being "not yet twenty", suggesting that they are nineteen about to turn twenty. They're still teenagers--little more than boys, arguably, though prematurely burdened with hard and gritty men's work, and about to be thrust into an explosively complex world of adult emotion and choices. Their youth adds to the poignancy of their dilemma.
Jack shows an amazing capacity to love, and an abiding hunger to be loved. The wonderful image of the little dog in his coat is just one detail suggesting the generosity of his heart.