Wouldn't bother me, I compare it to breeds of dogs, I absolutely love golden and flat coated retrievers having had several goldies they all look the same but each has been very very different. So a horse that looks the same wouldn't bother me.
I wouldn't give a new horse a name I had used before for another animal either, names are unique and the reference point in my mind for that animal so I would never use it again.
Say I owned a Jack previously and then brought another horse who had been called that I would change it. I don't believe in the whole 'bad luck to change their name'.
My horses previous owner tried to change his once or twice but he flatly refused to answer to it, one day in desperation she shouted his old name and he trotted over. So he decided himself that was his name and he wasn't having another one.
Well I call him other things sometimes 'Gerroffmyfootyoulump' for example but he quite happily ignores me