If you are showing the horse or dog it should be registered but
If its a pet who care if the are registered or not
They have to be a good match for the owner
A registered horse can be more of a handful too
I don't care if people register horses or dogs or cats or parrots or they don't register them - I'm just saying without registration, you have no proof that it's purebred. And in order to be classified or advertised as such, you'd need it.