Are these the only forms? That is, (picking just one for example)
Io ho il proprio gatto (I have my own cat)
Tu hai il proprio gatto (You have your own cat)
Lui ha il proprio gatto (He has his own cat)
Noi abbiamo il proprio gatto (We have our own cat)
Voi avete il proprio gatto (You have your own cat)
Loro hanno il proprio gatto (They have their own cat)
Are those all correct?
Oh, I get that part, no issues there. But for example, "il mio, la mia, i miei, le mie" does reflect that the owner is 1st person singular and "il vostro, la vostra, i vostri, le vostre" does reflect that the owner is 2nd person plural. I was wondering if "propri*" followed the possessive rule (also indicating whether the owner was 1st, 2nd, or 3rd person) or the ordinary adjective rule (matching only the thing possessed in gender and number). It looks like it follows the ordinary adjective rule. Which means that there's some basic context required to fill in who, generally, the owner is.
They are different forms of the verb 'have'. There are lots of irregular verbs in Italian, and another example is 'vanno' (they go) and 'andiamo' (we go).
It's a difference in reference. "Lui ha i suoi gatti", or "He has his/her cats", could mean he has anyone's cats. "Lui ha i propri gatti" can only mean the cats belong to the subject himself. It's also a matter of emphasis: "He has his cats" vs "He has his own cats."
He has her cats while she's away.
He has his cats to keep him company.
He has his own cats, so he doesn't need hers, too.
Io ho i propri gatti = I have my own cats.
Tu hai i propri gatti = You have your own cats.
Lui ha i propri gatti = He has his own cats.
Lei ha i propri gatti = She has her own cats.
Noi abbiamo i propri gatti = We have our own cats.
Voi avete i propri gatti = Y'all have your own cats.
Loro hanno i propri gatti = They have their own cats.
"He has his cats/Lui ha i suoi gatti" vs "He has his own cats/Lui ha i propri gatti" is a matter of disambiguating whether John has Oliver's cats or John has John's cats.
"He has his cats with him" specifies that the cats are explicitly in the same location as John.
Ha i suoi gatti con lui. ~ He has his cats with him.
Ha i suoi gatti con sé. ~ He has his cats with himself.