Human/mammalian immune system do not react to everything foreign that enters the body. This foreign substances are called "antigens". Instead, the host [i.e. body] reacts to "immunogens" - these are antigens, that CAN elicit immune reaction. Microbes are immunogens, for instance. They don't only consist of microbe proteins, but also of so-called PAMPs, pathogen-associated microbial particles. This can be a bacterial cell wall or "peculiar" DNA and double-stranded RNA, that are definitely cannot be found in mammalian tissue. And these PAMPs are recognized by PRRs, pattern recognition receptors, that are found on innate immune cells, such as dendritic cells and macrophages. WITHOUT activation of innate immune cells, without, basically, INFLAMMATION, the overall immune reaction is not possible. At least, this is a current concept.
So, horse antibodies are not sufficient to elicit immune reaction.
In order to induce immune reaction, these antibodies should be injected with something, that can activate PRRs or cause tissue damage, like aluminum dioxide. The latter substance is largely used in formulation of vaccines. Guess, why ;-)
Of course, there are immune reactions to proteins or harmless xeno-antibodies, but this is usually not physiological, if the manifested reaction is severe. Example: food allergy to egg protein, ovalbumin.
Hope, it helped.