Horse showing, for the majority, is not about breed. There are certain breed sections for owners to show their horses in, but the majority is based on the hours of training spent by the horse and rider TEAM. When you spend endless hours training for soccer, or football, or baseball, or ANY sport, you want to see a good outcome in competition. Your team includes other people.
For equestrians, the horse is your partner. After riding for hours every day, doing mundane things like working on collection or riding a line quietly (not galloping over hills like a cowboy), some people like to see if their abilities, not how beautiful their horse is, match people in their same riding demographic.
It is a sport. Horse showing is not a parade of pretty ponies in sparkly saddles.
And really now, you clearly have never had a personal relationship with a horse and then take them to a show environment — as one person said, they thrive. They absolutely open up. My mare Lizzie would turn into this beautiful version of herself, where she, not I, was proud of herself. When I got this mare from an owner who mistreated her, she would have behaved like the prey animal she is. She was skittish, flighty, and would have run for cover at the sign of anything unusual (just like a piggy). My handling calmed her down.
As far as the "breeding" aspect of showing you are thinking of, it derives from when horses were work animals and, like sarah0712 said, work animals needed to have certain conformational attributes. They needed strong legs, powerful haunches and a developed topline. Work horses made money, money kept families alive, families keep the world turning.
This "jumping" you're thinking of derives from field hunting, where horses were used as a means of acquiring food. It was also a practice in military training, where horses were used as a means of defense.
Food and defense.. those are pretty important to human survival.
Now, we have the technology and the use of horses have diminished to showing, where we continue the traditions laid down by humans before us. If you were to go into a top show barn, you wouldn't see the environments of show cavies and rabbits. You would see happy, sleek horses in stalls deeply bedded, full of hay. Most would probably have an attached paddock. Each horse is fed a highly specified diet. Each is groomed and handled daily. Each is exercised daily.
I will agree that it is highly different from cavy showing. I will not agree that it is inhumane.
hmy: There are even more homeless doggies out there than guinea pigs!
