I suppose most of it is pretty obvious - the subduing of a large creature that is about ten times more powerful than you; the friendship and adventure shared with it after it has been subdued; the horse as a form of transport to take you wherever you choose to go, and - possibly the most important - the overwhelming sense of legitimate power that is bestowed upon the (arguably) weakest and most vulnerable humans in society. There are few situations that carry more responsibility for girl than being placed on the back of a horse, and allowed to wander off alone through treacherous countryside.
I don't want to dwell on the 'Freudian' nature of the whole business (that is too simple and probably more of a figment of other people's lurid imagination) other than to say that it is mainly females who use hacking stables, though not exclusively, of course. You might just as well attach the same sort of connotations to the other sport of shooting, as it is mainly done by males (though also not exclusively), and there are all sorts of simplistic, Freudian interpretations that you could impose on boys and guns, but - once again - I think they would be more to do with personal power than anything else.
I have a woman friend who was brought up here in Bath, and she told me that - when she was about 16 - she was out horse riding with a friend, when a man appeared from out of the undergrowth, and asked them if they would care to take part in an experiment for a well-known watchmaker. He laid out a neat row of wristwatches on the path, and asked the two girls to ride their horses over them to see how the watches withstood the hooves of the horses. (!) A few years later, I was talking to another woman friend (who was not connected to the other) and she recounted the same experience when out riding in the same area, so this bloke was obviously a regular in the vicinity. I suppose - as far as these little foibles go - he was almost harmless, but, my god, what a fetish!
Now, dressage - that's taking it all to another level.