and what role does the market index play in the global financial system?
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The reason each country has different index and the reason for indices differ among countries is due to the way indices are constructed. Index of each country (in specific each exchange, because a country can have more than one index. For Eg; In india two popular indices are nifty and sensex) is constructed using the stocks traded in particular country/exchange. So each exchange will end having a unique index unless it is proxy to another one.
For example S&P CNX Nifty of India has 50 stocks in the index where as sensex of bseindia has only 30 stocks out of that. You can see the market indices in the same country differ depending on the stocks included in the index construction
Market indices differ among countries because every country has it’s own economic growth propective. Today, in global economy, the market index doesn’t seem to have a big effect on another.