A question for the native speakers (English, American): Why isn't if "of this country"?
Technically that makes sense too but you'd sound like a foreigner if you said that. Sometimes there is no reasonable explanation as to why we commonly say things one way instead of another. You just have to learn it without understanding why. I personally think the best way to say it is "I have this country's passport."
As a native speaker of English, I responded with "of this country," and it was considered incorrect.