1-- ' Of ' needn't mean possessive ('owned by'). Here, of is "used to indicate specific identity or a particular item within a category: the city of Chicago; thoughts of love . 2-- ' People ' here is, I believe, a bit of Marxist terminology: "used in the possessive in Communist or left-wing countries to indicate that an institution operates under the control of or for the benefit of the people, esp.
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It (People's Republic of China) is a name- they don't have to make sense in English usage and grammar just like in all other languages. "People's Republic of China" is just a name to used for propaganda- People's to just it for all people not a just the few as the country is in reality. The Republic of China is a not a republic- it is an oligarchy.