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It seems like the US is still the Wild West when it comes to public traffic laws. Here they wouldn't let you go or probably you will be asked for an unique driving license arguing that horses, unlike vehicles, are sentient beings who might cause an accident if they get distressed.
Personally I would like to own a horse, specifically a mare because I heard that they are useful not only to move from one point to another, but you can have fun in other ways.