Finally, I have my own car.
I dimly remember that “propria” (proprio …) already includes the possesive pronoun like in “ognuna invita la propria figlia”. (Yeah, from the dark side of language learning … 
)
It is here to stress the possesion more?
Proprio is a fascinating little word with so many uses. Come in Mike-lima;-) 
Montalbano the other night said “Non proprio” (not really) which I hadn’t heard before.
A dopo…
“Proprio” shares the origin with the English word “Proper”, but it has many more uses compared to it.
As an adjective, it can be used as a substitute for a possessive, but in this case, it means “private” or “exclusive” (think “property”). So “my own” is a fitting translation.
It may mean “proper” as well, (as in “proper behavior”, “comportamento proprio”, although “appropriato” is much more common for this case).
As an adverb, it means “exactly”, And it is used in a lot of common expressions:
“Proprio cosi’” → “Just so”, “that’s right”
“Non proprio” → “Not exactly”/“Not really”, but “Proprio no!” means “Not at all!” / “No way”.
Una spiegazione superba! Grazie come sempre.