The "mock breeding" is this: By Irish Peasant out of Bog Oak. Breeding refers to the parentage of the horse, which should be the two breeds of horses. I don't know horse breeds, but they are definitely not "Irish Peasant" (a person) and "Bog Oak" (a tree).

It means that "Irish Peasant" is the name of the toy horse's imaginary father, and "Bog Oak" is the name of its imaginary mother. "By ___ out of ___" is a commonly used phrase when discussing pedigree, or breeding, of racehorses. Of course, a toy doesn't actually have parents, so Brat made up an imaginary, or mock,pedigree for it.