You know, I actually do think that horse can feel attachment to people (nice people anyway). I don't have enough experience/knowledge yet to confirm whether that's true.
But I definitely think my first lesson horse has an attachment to me. Whenever she is in the pasture and I have to get her, I call her name and she saunters over to me. I like to think she does that because its me lol.
She is a quarter horse mare and although I've been told she doesn't have the best conformation for jumping, I would definitely buy her when I can afford to. Other horses I've ridden just haven't given me the same vibe she does. My new lesson horses are great too but there's just something special about the first horse who taught you how to ride.