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Calmette tried injecting horses with snake venom and then taking out the serum which has produced antibodies against the venom and injecting in the snake-bitten human.

Shouldn't our immune system recognize this exogenous protein and try to eliminate it as it's a non-self protein?

Our immune system does react to horse antibodies, but as with any adaptive immune response it takes some time for the response to develop. In the weeks before our immune response fully responds to the horse antibodies, the infused antibodies can have their effect.

If you then have a second infusion of horse antibodies, the immune response would not only prevent them from having their beneficial effect, but it might case serious adverse effects. This serum sickness (Wikipedia link) has been known for over a century; here is a paper from 1918 discussing it as a well-known issue:

1. The injection of horse serum either in small or in large amounts in human beings is always followed sooner or later by the development of hypersensitiveness of the skin to subsequent injections of horse serum. For the development of this reaction serum disease is not essential.

--THE RELATION OF CIRCULATING ANTIBODIES TO SERUM DISEASE. J. Exp. Med. Published March 1, 1918

Although there aren't many modern situations in which horse (or other non-human species) serum is injected into humans, when it is (as with anti-venom, which is generally produced in non-human animals) serum sickness is still a concern.

Serum sickness is a delayed immune reaction resulting from the injection of foreign protein or serum. Antivenom is known to cause serum sickness but the incidence and characteristics are poorly defined. ... The primary outcome was the proportion with serum sickness, pre-defined as three or more of: fever, erythematous rash/urticaria, myalgia/arthralgia, headache, malaise, nausea/vomiting 5–20 days post-antivenom.

--Incidence of serum sickness after the administration of Australian snake antivenom (ASP-22). Clinical Toxicology. Published 2016

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.

Best,
S