They hold shared on behalf of other people, i.e. if someone buys one of their index funds they buy the corresponding shares. They are the on-paper owner of a very large fraction of shares in public companies, but they are a) not the ultimate owners, just agents of the real investors, and b) as a consequence very hands-off in the operation of the businessness. They in general just go along with what the company executives and other, more direct investors want. They have a general stated goal of 'encouraging long-term value for their stockholders', but the most activist thing they've done was contribute to a shuffle in exxonmobile which pushed them to pay a little bit more attention to the environment and climate change, which is if anything the opposite of what people tend to assume they do.
I mean they also send me an email to vote with my fractional S&P 500 shares whenever a shareholder vote comes up, so even if they're the on-paper owner of the shares they seem to pass through the voting rights every bit as much as the direct value and dividends.
You are probably voting for the ownership of the fund, and not the companies the fund owns. Unless you get around 500 different share holder vote forms every year you are not voting for the companies in question you are just voting for the leaders of your funds. (around 500 because S&P 500 funds often buy companies like what is in the S&P500, but not always exactly the same companies. Even if they want to be exactly the same companies they take time to buy and sell anytime the S&P500 changes just because the market could not handle them buying/selling everything the minute the S&P list changes)
