The event to which you are undoubtedly referring happens in chapter two of Anna Sewell's Black Beauty. The horse narrator of the story, here named Darkie, is two years old when he sees a terrible thing happen to both a horse and a man.
Darkie is in a corral with his mother and some other colts when they see a hare race by, followed by some hunting dogs and then their owners on horseback. The hunters jump their horses rather recklessly, and in a flash two of the men and their horses are down. Only one of the men gets up; the other has a broken neck and is dead.
The dead rider is the "young George Gordon, the squire's only son, a fine, tall young man, and the pride of his family." While others run to get the squire and a doctor to care for the boy, the farrier quietly examines the horse, Rob Roy.
When Mr. Bond, the farrier, came to look at the black horse that lay groaning on the grass, he felt him all over, and shook his head; one of his legs was broken. Then some one ran to our master's house and came back with a gun; presently there was a loud bang and a dreadful shriek, and then all was still; the black horse moved no more.
In contrast, the funeral for the Gordon boy is, of course, much more emotional.
Not many days after we heard the church-bell tolling for a long time, and looking over the gate we saw a long, strange black coach that was covered with black cloth and was drawn by black horses; after that came another and another and another, and all were black, while the bell kept tolling, tolling. They were carrying young Gordon to the churchyard to bury him. He would never ride again. What they did with Rob Roy I never knew; but 'twas all for one little hare.
Your question implies that readers should be offended by the disparity between the treatment given to Rob Roy and the treatment given to George Gordon, and it is true that none of would ever put up with a doctor just shooting a human because he broke a leg. It is also true, however, that the farrier gave the horse a thorough examination and was not happy about having to shoot the horse.
What is most striking about this incident to me is that the mourners act the same. After Rob Roy dies, Darkie's mother is "much troubled" at Rob Roy's death, especially because she feels as if the Gordon boy was being a bit reckless while he hunted. "[H]e was a good horse, and there was no vice in him. She never would go to that part of the field afterward."
While the story does not say so, the Gordon family probably feels the same way about their son as Darkie's mother feels about her fellow horse, and surely they will feel grief every time they see the spot where their son died. The treatment may be different (which is not particularly surprising), but the reaction by the loved ones is nearly the same.