Can I ask - what's up with horses? Here in the Russian, and in the German lessons, too. Everyone has a freakin' horse, cat or a dog! Why doesn't anyone have a ferret, or a hedgehog, or a parakeet, etc. It just seems like a missed opportunity to squeeze in some more vocabulary about animals. Just sayin'...
The student's gender is clear since it's студентка so there is no need to use the neutral pronoun. On the other hand accepting "their" would cause confusion and create ambiguity which doesn't exist in the Russian sentence. "This student loves their horse" sounds as if you are talking about somebody else's horse ("Эта студентка любит их лошадь"). The Russian sentence clearly talks about her own horse.
свою refers back to the subject of the sentence, so in this case will mean 'her', but is agreeing with the object, hence it's in the feminine accusative singular case (лошадь is also in the fem. acc. sing. but fem. nouns ending in a soft sign don't decline in the acc. sing.)