Yes, it is correct.
The third person singular pronouns "he", "she" and "it" usually distinguish between antecedents which refer to human beings, and antecedents which refer to things which are not human. The pronouns he and she usually refer to human beings. However, "he" or "she" may also be used to refer to an animal, if the gender of the animal is known to the speaker and / or if the animal is a pet [ and it is considered as a "member" of the family]. The horse [in your example ] is the main character on which the story is centered; i.e. "he" is a specific animal not an ordinary one in this story.
e.g. My aunt is very intelligent. She has a degree in Philosophy.
His brother was not there, because he had to work late.
Your dog is well trained. He is a good guard dog.
In the first two examples, "he" and "she" are used to refer to human beings. In the third example, "he" is used to refer to an animal.
If a third person singular antecedent does not refer to a human being, the pronoun "it" is usually used. However, when the thing named by the antecedent is referred to as if it had human qualities, the pronoun "he" or "she" may be used. For instance, ships and countries are sometimes referred to as "she".
e.g. There is a robin on the lawn. It has a nest in that tree.
This is a good book. It is interesting and well-written.
The Bonnie Belle is a sailing ship. She is very seaworthy.
In the first two examples, it is used to refer to things which are not human. In the third example, she is used to refer to a thing which is not human.
Ships and countries [Source : Wikipedia]
.... "Traditionally ships have been referred to using the feminine pronouns (even ships named after men, such as the USS Abraham Lincoln), as well as countries and oceans. The origins of this practice are not certain, and it is currently in decline (though more common for ships, particularly in nautical usage, than for countries). In modern English, it can be said that the use of the pronoun "she" to refer to inanimate objects is an optional figure of speech........"
I hope it helps.