I know that a stock market index (like S&P, NIKKEI, etc.) is computed from the prices of selected stocks, but I was wondering why the name of the currency is almost never expressed. For example, if the S&P price is 2927 points I assume those are dollars, but how about NIKKEI? Is its price is expressed in yen? Is DAX expressed in euro? Is there a standard in financial community?
Indexes don't have units. They're indexes.
An index is a ratio of two numbers. The two numbers have the same dimension. So their ratio is unit-less.
The compilers of each stock market index set a denominator which represents a base set of conditions, and they then may adjust that denominator to ensure that events that change market cap, but which should not change the index, do not change the index. The sum of the market capitalisations of all of the index members forms the numerator. These are both expressed in the domestic currency (so for the FTSE 100, the currency is GBP; for the S&P 500, the currency is USD)
Stock market indices are still exposed to currency variations, because the denominator is not adjusted for currency variations, and the numerator is exposed to them: the index members' market caps are all in the same currency, and their profitability will almost always have some exposure to the currency too.
For a multi-currency index, a standard currency is chosen, and then each company's market cap is converted to that currency. Again, the numerator and denominator are expressed in the same currency, so the index has no unit.
The one freak in all this is the Dow Jones Industrial Average, which doesn't behave like other indexes (it's an index, but it's a straight price average, not weighted by market cap), and should, broadly speaking, just be ignored.
While stock indexes can be formally defined as unitless, they are tied to a currency because comparing market values at different times requires a specific, persistent unit of account.
A simple criterion that clarifies the issue is to ask in what currency a given index is investable, i.e., in what currency an ETF could be traded such that its price maintains a nearly fixed ratio to the index. For national stock indexes, this is indeed the local currency: dollars for the S&P 500 (and, yes, the DJIA too), yen for the Nikkei, euros for the DAX, etc.
Thus, you will find that US-based ETFs for US indexes track the indexes directly, whereas US-based ETFs for non-US indexes track the indexes multiplied by the exchange rate of the index currency relative to the dollar.
Global indexes such as MSCI World have to specify their choice of currency and may quote the index value separately in dollars, euros, etc., with different numerical values and percent returns.
Again, this is a unit of account issue, as the economic substance of a stock investment is not affected by the currency in which one measures it.