Those bashkir curly horses are indeed hypoallergenic, making them pretty much the equine equivalent of poodles. They even have the same curly hair. It’s quite funny!
As for the donkey allergy, I’m not sure. Horses and donkeys aren’t the same species, but share the same equidae family. If they were the same species, donkey and horse offspring would be fertile. Sadly however, mules cannot breed. Donkeys have 62 chromosomes in 31 pairs. Horses have 64, held in 32 pairs. The breeding between the two of them creates a creature with 63 chromosomes. And an odd number of chromosomes cannot be split to make offspring. Because of this, science would suggest that horses and donkeys are not the same thing, but are related enough to produce healthy offspring. Even if it is infertile.
I’m not an expert on allergens, but I’m guessing that you might be allergic to donkey as well. A google search yields results for milk allergens in both donkey and horse milk, but nothing about hair. It might be wise to ask a medical professional for the answer, and to test.