Cross entropy is identical to the KL divergence plus entropy of target distribution. KL equals to zero when the two distributions are the same, which seems more intuitive to me than the entropy of the target distribution, which is what cross entropy is on a match.
I'm not saying there's more information in one of the other except that a human view may find a zero more intuitive than a positive. Of course, one usually uses a evaluative method to really see how well classification occurs. But is the choice of cross entropy over KL historic?