Honestly, I suspect that Strife, War and Fury will mock Death mercilessly about his affection for his human 'pet' but only when away from her.

There is a certain front of dignity to maintain in front of mortals when one is a Horseman, even if said mortal is seen you on Laundry Day and watched your utterly pathetic reaction to the threat of intravenous anaesthesia ("NO NEEDLES!"), bugs ("FILTHY INSECT! DIE!") and paper-cuts. So, in front of her, there is to be no hint that Death has tender emotions and sees her as anything but a rare adornment to parade before the Council's lesser agents when they gather.

Because tough guys tend to have neuroses like that.