While at the first uneducated glance this solution seems viable (as many other solutions do), it has several underwater problems that render it unusable or at least less efficient than solutions it was designed to replace. Just a short non complete list: 1. You have to feed the horse even if you don't drive, which may eliminate any economy - depending of the use frequency 2. Horse excrements collection not solved 3. Most people do not have the place for horse to live etc.
To Misha's list, I would add the question of why they've got the horse *pushing* the car instead of pulling it. A good horse can pull up to six times his own body weight (though they're generally worked far lower, at more like 1 1/2 times per horse) but pushing?