tl;dr: horse blood isn’t special, humans are special, so they use the horse blood because it’s less bad to hurt a horse than a human (also possible the disease in question doesn’t hurt horses as much as humans but I’d guess mostly the first one).

Human blood creates antibodies for every disease in our body as soon as we get it too. There are two reasons scientists use horse blood for this and not human blood: 1) injecting the disease into human A so that human A produces antibodies against it that will protect human A defeats the whole purpose - the reason we get sick is that it takes some time for the body to figure out the right antibodies and ramp up their production. So deliberately injecting a disease-causing agent just… causes the disease, and presumably it’s a disease with unacceptable odds of hurting or killing you while the antibodies get made otherwise why would you bother with this, you’d just passively wait to catch it. Like with colds. And 2) injecting the disease into human A so that they produce antibodies that you’ll then inject into human B would effectively protect human B, but at the cost of human A getting the disease which becomes straight-up Mengele evil at scale. Having said that I think there were COVID treatments that were basically this, just without the evil part of deliberately infecting people.

Also, kind of besides the point given the rest of the answer but for educational purposes, injecting horse blood into humans would be harmful because it’s a complex liquid chock-full of living cells and compounds that the human immune system will recognize as foreign contaminants and attack. And if there’s enough of it (and you seem to be imagining a pretty significant transfusion here) this attack will be severe enough to harm or even kill the human in question. It’s the exact same reason you need to be careful with blood types when transfusing from human to human.