Why does it seem like more women than men like horses and horseback riding?

It seems this way because, in this savage age, unfortunately, it simply is this way – certainly in the US and UK. The more interesting question is how it came to be so. On the face of it, it is paradoxical: high school dressage, at core, is a martial art, originating during a time when matters of honor (among men) were settled via mounted 1:1 duels. The advanced horsemanship and training required for this endeavor eventually became an art unto itself, as well described in the two manuals written by the Duke of Newcastle during the 17th century. The practice of the mounted duel was outlawed, and with the advent of gunpowder and long range weapons, the role of the horse in the military had been reduced to that of a troop transport by WWI – until the widespread use of the automobile, which overtook that role as well. Since then, the pursuit of high school horsemanship has been left to the very few with a passion for it, the time for it, and the means to pursue it, while also affording the expense of one or more horses, which, on average, probably equates to an extra rent or mortgage payment. Thus, there is a bias towards the non-working and wealthy – you can do the math from there. Moreover, it is an artistic endeavor, so, as with most of the fine arts, if you pursue it, you can either be a female, else be a male whom people will suppose is gay – which is, of course fine, but it is an unfortunate stereotype which keeps too many men – sensitive to being mis-cast, as it were – away from an historically gentlemanly, noble – moreover, martial – pursuit.