When I fell out with the AES, the previous owner of the company (Henk M) assured me that name changing on a whim was illegal. I phoned a Whitehall number, got a junior Minister's mate and asked him, he fluffed and guffed and said that the department had decided to leave it to the individual Breed societies.
It also seems that even that august body Wetherbys permit new owners to change a horse's name. I gather that it's an expensive procedure which may explain their acceptance!
I've decided that life's far to important for it to be taken seriously.