It does look like the nameplate was added on after the original order - especially considering the digs above the nameplate - as if there was another one before which was removed. And if you hadn’t mentioned your mother receiving it from the Hancocks, I’d question it’s authenticity. In antiques sales, have seen many added/switched labels. And can’t imagine they would have put it on upside down from the factory.
But since it came from the Hancocks. maybe it was worn by another that was around, became an extra at the barn, and in an emergency one day they had to use it for Secretariat? And it fit him so well, so well weathered - they just ordered the nameplate and someone at the barn put the new nameplate on upside down on this other horse’s halter?
