Firstly, the constructions of your sentences are different in Turkish.
Boris has a dog: Boris'in (bir) köpeği var.
Boris has the dog: Köpek Boris'te.
So there isn't any accusative here. But if a word contained both acussative and possessive marker, possessive would come first.
Köpeğini buraya getirdim: I brought her dog here. (köpek(dog)-i(third singular person possessive)-n(buffer letter)-i(accusative))
Babamı ziyaret edeceğim: I will visit my father. (baba(father)-m(first singular person possessive)-ı(accusative))
Konuşurlarken seslerini duydum: I heard their voices while they were talking. (ses(voice)-leri(third plural person possessive)-n(buffer letter)-i(accusative))
Onun saatini çalmışlar: They stole his watch. (saat(watch/clock)-i(third singular person possessive)-n(buffer letter)-i(accusative))
This sentence is more like 'There is a dog which is Barış's.' The part indicating that it is Barış's is the '-i' third person possessive suffix there. Because 'var' doesn't give us the possesor.
Barış'-ın(genitive) bir(a) köpek(dog)-i(third person singular possessive) var(exists/there is).
Yes, I think this is it. Genitive means that someone or something belongs to someone or to something. I cannot find any exceptions, maybe there are, I don't know, I mean possession in a very wide sense. So Bariş'in is both here (???). The possessive pronoun replaces the owner, so why use both? It is a general notice, valid for many many languages. In English is easy to find, some others use declension, including Turkish. In Romance languages I know comes from Latin De=about, etc. But here? Is it a genitive or possessive?