Death is a bit like Data from Star Trek TNG - the more he interacts with humans the more human-like he becomes. Humanity has become an interest to him and that interest has become contagious. We see him in the various books trying to understand human emotions and the human way of thinking because humans are his job.
I think there's something in one of the books that says something like a rat catcher will, as he learns about the rats he kills, become fascinated in them and eventually may even become a little rat-like. It goes beyond a job and becomes an interest and then a passion.
If I remember correctly, when the rat catcher in Maskerade dies, the Death of Rats comes for him.
So Death becomes more human-like as the books progress and he has more contact and takes a greater interest in humans.