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Christianity is not, "just a belief system", nor is it "also a system for organizing and governing society".
It is simply the way for imperfect people to get into an eternal relationship with a perfect God, not by our good works but by the gift of Jesus. John puts it better than I can:
For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.
Of course, there are consequences of this that affect the individual (e.g. a commitment to honesty and love) which then in turn affect society. (We are also instructed about our duty to society, e.g. "pay your taxes".) But that's a by-product, rather than the central aim.
More directly answering your question, the Bible does not explain how to organise society but it does contain some very light weight (by which I mean, not overly proscriptive) examples of how to organise a church (e.g. mentioning elders).
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If you liked Mere Christianity, you might like to read Screwtape Letters and see what Lewis has to say about using Christianity to further political ends.
That being said, Christianity is the basis for a Church, which is a bigger thing than a government. In the OT, we've got the Covenants (Individual: Adam and Eve, Familial: Noah and his Family; Tribal: Abraham; National: Moses; Kingdom: David) and in the New Testament, Jesus establishes a new covenant with His Church, one that will not pass away.
If the old covenants were used to organize and govern those societal groups, getting bigger and bigger until the Kingdom of Israel. Then what Christianity is, is actually too big to be used as a tool to govern a society, it's for all people, even though not everyone accepts it.
This is why at consecration, the priest says "for many" in the English translation now, they had it translated "for all" for the last 50 years, but Christianity is really just for those who accept Christ.
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Islam proposes a system of government. But the Bible does not. Although many systems with Christian beliefs have tried to force a political system or government. I would say that they failed.
I am not saying in any way that a Christian should not do politics and act out of his belief system. What I am saying is more that there is no such thing as a Christian nation but there is an Islamic nation.