A share market and a stock market are terms that are often used interchangeably. In India, we often come across investors using the term share market and not stock market.
In countries, particularly in the West, the term that is more often used is stock market. This has nothing to do with the instrument that one is holding and whether it is a stock or a share, as both are the same.
So, if you want to say you own the shares of "Reliance Industries" you would not say, "I own the stock of Reliance Industries".
So, essentially, there is a marginal difference between a share market and a stock market. Individuals, also often use the term equity market as well. This is more often linked due to the equity class of shares. In the good old days, there were many types of shares that were issued, including equity shares, preference shares and shares that were converted from convertible debentures to shares.
Hence equity market came into vogue because of the equity class of shares. There is also another class of shares with differential voting rights called DVRs. Companies like Tata Motors have issued these class of shares.