Do horses get emotionally attached to their owners like dogs?
Yes, they do. Very much so. And they have long memories for both the humans they've bonded with in a positive way and the ones who have damaged or abused or frightened them.
The depth of the connection depends greatly on several things, not the least of which is the amount of time the human spends with the animal. I've had the experience of boarding my horses at someone else's farm where I saw them for anywhere from an hour a day to once or twice a week, depending on circumstances. My horses knew me and responded to me with a certain level of connection, and I repaid the favor. But that was nothing compared to having them here on my own farm for the past 19 years. They know my routine by heart, know me as a friend and companion, and seem to care about whether or not I show up.
I'm not a believer in the "joining the herd"' theory. Humans are not part of the horses' herd. We are something else that isn't as easy to name. We are their source of food, comfort, and safety if we are good owners. We are their source of pain and anxiety if we are not. If we ride them as well as simply owning them, then the connection becomes stronger through the body contact and we form a coworker relationship with them. It's just not as simple as it sounds.
The depth of relationship also depends greatly on how well-informed the human in this scenario is. I've known people to own horses for decades without one iota of understanding of how their horses think, what they're feeling at any given time, or what their needs and desires might be. They have no explanation for much of their horses' behavior and frequently do things that make me shake my head. Those horses seemed happy to see anyone who had an inkling of how to relate to them, but no real connection to their owners.
Horses communicate almost entirely through body language. We humans never shut up. We are verbal to an annoying degree. If the human has an interest in a true bond with the horse, then silence becomes his best approach. Low voices, nickery sounds, cooing and silence are the staples of a horse's life. Horses are smart enough to learn voice cues, and that sometimes misleads humans to believe that we need not learn their language. But we do need to do that. Simply controlling eye contact is huge in the horse world and can be put to use in causing a horse to appreciate the time with your or hate it. There will be a closer bond with humans who understand the language and social structure.
I know for a fact that the capacity for deep bonding is there. Call it love if you want. It's a connection of some psychological type. I know of a man who sold the horses who lived in his back yard and had them jump the fence at their new home and follow his truck down the road until he stopped and picked them up and returned them. Is that love, or is that dependency and anxiety? It's impossible to tell, but it certainly is connection. I know of a horse that stopped eating when his young owner bought a new horse. His jealousy was so strong that he lapsed into depression and had to be sold to a new owner at a different location before he was able to recover. He was certainly attached to her. One of my horses took it very personally that I'd decided to take one of his herd mates out for lessons off the farm, and he attacked and bit the other horse whenever I went to get him from the pasture. Jealousy has to come from a desire to be the one in the situation and anger at not being that one. That suggests some sort of bond. And when my own daughter once went on a two-week vacation and left her horse home, he pined for her attention. He perked up when I took over riding him, but his joy when she returned was visible.
On the other hand, my two most closely-bonded horses sulked for two weeks when I brought them back to the farm after a week at the State Fair where they'd apparently enjoyed the excitement and the attention of strangers. Go figure.
So the answer is a "yes, with reservations".