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.