A horse is not a pet. Here's why:
1) A pet loves you without any purpose behind it. If you don't know what you're doing with a horse, it won't be kind to you. When I first got my horse, I will ill-advised to use a shank bit to control him and a tie down to stop him from slinging his head around. I had no idea what I was doing but he let me know that it was very unpleasant by throwing me onto hard dirt and rock which permanently injured my elbow. After learning the mistakes of my ways, I was trained in horsemanship and had my horse trained. Now I can ride him using no tack but a halter and lead rope.
2) A pet is completely domesticated, but a horse isn't. A horse is a wild animal that acts upon instinct. Even after having one trained, the horse will still have wild instincts such as an intense fight or flight response to anything that scares it or angers it. In order to have a horse, you have to understand how to recognize and respond to those signs of anger or fear before the horse takes off with you, throws you, or kicks another creature. Having a horse requires a great deal more understanding than a dog or cat would.
3) A pet learns after getting in trouble that they should not misbehave without intense disciplinary action. A horse is like a teenager with a bad attitude problem. If you don't give them reason to respect you, they will quite literally walk all over you. I'm not saying to abuse your horse, but it takes working the horse until you are sweating profusely with aching muscles before it will usually learn its lesson. You can't just spritz the horse with a spray bottle filled with water and expect it to listen to you.
4) A pet doesn't really reflect you unless you just coddle your pet without teaching it manners. A horse is as good as its rider, therefore the way your horse acts reflects your stature as a rider and/or trainer. You are a team. There must be a mutual respect present in order to really bond with your horse and become a team.
A horse isn't a pet, but it is a part of your family. It's a part of you and it teaches you just as much as you teach it.
