I'm not a native Italian speaker, but if it's anything like it is in Spanish, my native language, then I could help. I might be wrong though, so take it with a grain of salt.
Ok, here it goes.
If you say "la zuppa non è LA mia" it implies that you do own a soup, but the one you're referring to is not yours. I understand it as if you were looking for a specific soup that you had previously identified as yours, and then you find one and realize it's not THE one you were looking for. So it's like saying "The soup is not the one that belongs to me" or something along those lines.
In contrast, if you say "il cavallo non è mio" (without the article "il" before "mio") you're simply saying that the horse is not yours, but it is unknown if you have one or not. It's just not yours.
Does that make any sense?
Yeah, I'm a mostly fluent Spanish speaker and that's how I assumed it worked as well, but I don't think so. My understanding now is that in Italian you absolutely need an article with possessive pronouns unless you're dealing with a) family or b) predicate nouns, in which case the article becomes optional.
Probably because "that" is the subject word of your sentence, so basically that sentence could be said if someone put a cat in your face and asked if it was your horse, whereas the sentence for the question is more so if someone showed you a horse and it was not yours, hope that helped clear things up! :)
But let's look at it from different scenarios, scenario one. You are called by your friend who says they have found a horse, and they are wondering if it is yours, in reply you would say "the horse is not mine". Second scenario, you are out and about looking for your lost horse with your friend and they come back with a cat, you would say "it's not my horse". Also please take into account that Italian is not English, so their way of saying things is very different from English. So don't get mad at Duolingo, people took time and dedication to create these courses for you. Please understand.