Well, here's the deal,
You're a young guy, so I sense what you're really asking is "If I'm a cowboy, will girls like me?" Here's the trick.
Girls and women are not a monolith, and that's a good thing. You'd be surprised how many guys out there can't get that through their heads. Girls are people just like boys are people, and lots of different people like lots of different things. No matter what kind of guy you are, nerd, business guy, lawyer, cowboy, surfer, fireman, sports fan etc, there's a girl out there that probably thinks that's hot. Which is awesome.
If you're trying to find your identity and think "Guy who likes horses and rides" might fit into that identity enough to claim some sort of "cowboy" cred. Here's what you should keep in mind.
Girls have a fantasy about cowboys, and boys have a fantasy about cowboys. It is not wise to confuse the two and become the boys fantasy about cowboys.
Here's the difference. These are the things that are good about cowboys that most women I've met would agree are swoon-worthy qualities.
Integrity- A cowboys does what he says he's going to do and he does it with a sense of honor. A cowboy keeps his word.
Humility- A cowboy knows every once in a while a horse can and will put him in the dirt. These things happen so it's best to accept such faults and do your best in spite of them.
Work Ethic- Cowboys don't whine about things being difficult. They get the job done, no matter what that job takes.
Chivalry- Cowboys talk nice to their mamas, and don't mind opening the occasional door for a lady.
Manners- Cowboys know how to say , "Yes ma'am" and make it sound charming.
Care- Cowboys care for others, whether its the animals they work with or the people who are around them. Cowboys are there to help.
Resourcefulness- Give him a roll of duct tape and a stick, and then watch in awe.
Confidence- A cowboy never looks at his shoes and says, "Well, I don't know, I guess so."
If you asked me what was sexy about the cowboy mystique, that would be it, combined with a fantasy about a man that can cook over open fire while camping and going riding with me.
The problem is, the image of cowboys through the male lens is much different. It's rough and tough and no holds barred. Shoot em first, ask questions later, and get back in the kitchen and bring me a sammich, little lady. (Yeah, no.)
That's why you get these "cowboys" who are all cocky and full of it thinking that they're being some male fantasy come to life. They just chose the wrong fantasy, and now they're jerks. Don't be that guy.
I see nothing wrong in trying to find an identity though and attempting to parse out the goals of how to conduct yourself from the good that comes from that identity.
I'm all for it. Yay cowboys!