It's the singular vs plural & polite difference. Du refers to one close person you are speakibg to, can be a friend, can be family... Ni can be formal, as in when you are speaking with someone you are less acquinted with or has some kind of superiority over you from seniority, rank or any other reason. It is also used to refer to a plurality of people you are speaking directly to, like "you two should stop messing around and get to work". Its a really common structure really; french has it as Tu vs Vous, turkish has it as Sen vs Siz and even english used to have it as You vs Thou but they scrapped the latter in the modern english
Multiple people share one single horse. Or the speaker is being polite, and there's only one person who has one single horse. If there were multiple horses, one for each person involved it would be "ni har var sin häst" (you have one horse each) or "ni har alla en häst" (all of you have a horse).