If you only plan to invest in shares listed in your local stock market, then in most countries you’ll have plenty of brokers to choose from. If you want to invest in some foreign markets, you may have very limited options. Check the international stock broker guide to see which firms will trade which markets.
You can have accounts with more than one stock broker and many investors do. But it’s often convenient to try to get two or three comprehensive accounts covering most of your investments, rather than having to open a new account every time you want to buy a stock listed in a different country.