I do not have any doubt that horses have individual personalities. After you own and care for several horses you can form an opinion about this yourself. People who own dogs or cats agree that these animals have personalities. However, few people believe animals such as fish or reptiles have personalities. It depends on the intelligence of the animal and its ability to interact with humans. I believe that a big part of this is the animal's own interest in relating to people. If a horse approaches you and makes eye contact with you then you are relating to each other as individuals.
So what exactly do I mean by personality in a horse? Certainly, a part of this is a personality trait of gregariousness on the part of the animal towards humans. When speaking of personality the whole is greater than the sum of its parts. What is that magical quality of charisma that some people clearly have? I find charisma in humans much like Supreme Court Potter Steward's quotation about pornography. It is hard to define, but "I know it when I see it." For example, it seemed that actor Peter Ustinov always managed to steal a scene, even if his role was a minor one, solely by virtue of his charisma. Kennedy had it and Nixon did not. There are those horses that you see at shows that also have that hard to define quality of charisma about them. Do not mistake beauty for charisma: they are different. A beautiful person or a beautiful horse can come across as emotionally flat and lacking in personality. Charisma is not self-centered or self-possessed but is about reaching out and engaging others in a positive way. It cannot be forced although some of its skills may be learned. In humans it often lends leadership authority to the person possessing it. A charismatic person does not need to be on a mission but when one happens to have the ability to charm and influence others, it is tempting to use it to some end. Human charisma is both seductive and seducing and one needs to cautiously size up the motivation of a charismatic individual. Fortunately animals have rather transparent agendas.
In prior times horseman did not worry about an animal's personality. There was this sense of "breaking" a horse by forcibly bending it to your will. Today most of us feel these old techniques are unnecessary and often constitute a brutal bullying of an animal. Horse trainer Monty Roberts talks a lot about the old approach in his various books which are an appeal towards more humane and animal centered training methods. Understanding an animal's personality is helpful in your approach to the training process.
A big part of the human-equine bond is the issue of trust: you both need to trust each other. I need to have trust in this huge animal when I am up on its back and it needs to trust that I will treat it in a humane and caring way. The heart of the matter is not just the personality of the horse. It is as much about the personality of the rider as it is the personality of the horse.