Having 2 different names could definitely happen with two breed registries, as well. Most (all?) of them require every horse's name to be unique. For that reason, they often require you to submit multiple names when registering and they go down the list until they get to one that's not already taken. And of course, the registries don't coordinate with each other, so if you had a horse that was eligible for registration in both APHA and JC (as an example) you might submit the same list of names to both registries, but APHA might accept name #3 from the list, and JC might accept name #4. In that case, the horse would have 2 different names.
And as SR mentioned, USEF isn't a breed registry, they just register the horses to track their performance records. You can give whatever name you want to them and that's the name they'll associate with your horse.