Is this the mannequin's jacket? Yes, it is its jacket.
As I said, this is a mistake of Duolingo. They want "Is this your jacket?" but it is impossible to tell from a spoken sentence out of context the exact meaning. "Is is his/her/its jacket?" is correct, and "Is is your (formal) jacket?" is also correct. You can only get the exact meaning they are looking for if you saw it written: "È la Sua giacca?" Notice the uppercase S.
You are right. You can translation "la sua pittura" as its painting ("Questa pittura è del MOMA? Sì. È la sua pittura."), but "I drink coffee shop doesn't make sense in English and that meaning wouldn't occur to an Italian speaker if you said "io bevo caffé." I hope that helps. I am wondering if you are trying to understand or whether you are being argumentative.
You can write his or her as you like. But "sua" means his, her and its. It depends on the context. "Is this the computer's keyboard?" "Yes, it is its keyboard." "É la tastiera del computer?" "Sì. È la sua tastiera." It doesn't have to indicate ownership, only that it "belongs." So, in that manner, a mannequin can have a jacket.
Because sua means his/her/its as well as the formal Your. "Professore, ecco la Sua giacca." (Professor, here is your jacket.) Professore, è questa la Sua giacca?" (Professor, is this your jacket?) In written Italian, the formal version is often written with a capital S for clarity, but not necessarily.
This is from the days of manual typewriters, when the accent marks were too short to fit over uppercase letters. It had not been done in handwriting, and it no longer needs to be done in typing, although some people from certain generations still cling to it. (Warning: this is how a Spaniard explained it to me for Spanish, maybe things really were different in Italy.)