The other answers do a good job of detailing why the ownership of all shares of a stock must be kept track of.
However, I'm going to give a counterpoint: that there is no simple way for a company to track down who all its shareholders are in terms of actual people. There are at least two reasons for this:
- Shares are frequently held in a "street name". When you have a brokerage account with shares in it, the shares aren't actually held in your name unless you actually request the stock certificates. This is a convenience, as holding the actual certificates greatly increases transaction costs (you have to go through the transfer agent, there is extra time before you can buy and sell, etc.).
- Someone who owns a lot of stock - especially, but not necessarily for nefarious purposes such as trying to manipulate the price of the stock - likely isn't going to hold all of it in their own name. There are likely going to be lots of dummy corporations, entities and intermediaries. Trying to trace the actual ownership will be like those dramatic TV shows where they are trying to trace a call and the crosshairs keep jumping all over the map as you try to peel back each layer.
This isn't of course to say that you can't find out if a specific person or small set of persons own stock in a company, it just takes a lot of time and detective work.
Also relevant, someone trying to manipulate the price of the stock doesn't even need to buy or sell any shares of the stock itself. A big entity can transact in the options or futures market to obtain control of big chunks of stock without actually owning them. Some years back there was a big short squeeze on Volkswagen stock, when it was announced that someone had effectively gained control of enough shares to take over the company - but as I recall much of this was through private derivative agreements with banks.