Simple way is to create a random value for speed/coloration for each horse, and make breeding fast horses with other fast horses the only way to increase their speed. Give every user a random horse, but put a limit on how fast a starting horse can be. That way no one gets a perfect horse just by luck. Keep the stats hidden. The only way to know is to race.

More complicated option is to have values for each part of the horse randomized. Eye position, skeletal density, size, muscle strength, etc. Make all of those values contribute to different broad stats, and determine it via a basic genetics system. Then you would create different types of competitions that cater to specific types of horses. This is a bit gamey, but certain features might give a horse higher stamina, so you would use them for endurance riding. Others might have a higher focus stat so they would be better at show jumping. Then you have your basic races determined by speed. You could also have competitions for breeding a horse with specific colors/stat values within a certain amount of time. Whoever gets closest wins. There's a million other things you could do as well, like player ranches, using other dramanauts as jockeys, horse training etc etc.

Like I said before, the key would be to keep the raw stat numbers (endurance, focus, speed, etc) hidden. The only information people should have is the basic parts (eye pos, skeletal structure etc etc), not how those basic parts factor into the whole. You'd have to figure that part out for yourself. The only way to find if your horse is of good stock would be to race them, or know a lot about the inner workings of the system. Also, keep dramacoin out of it. Otherwise it'll just be a poweruser's game.