What does a stock market index stand for? I know that a stock market index (like S&amp;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&amp;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&amp;P 500, the currency is USD)

For more, see https://www.investopedia.com/ask/answers/040215/what-does-sp-500-index-measure-and-how-it-calculated.asp

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.