There are literally thousands of payment processing companies across the globe. As a merchant, you must know that all the payment processing companies do not work in all countries and with all industries.
Depending upon their licences, they operate in specific countries. A payment processor registered in the European Union may not provide services to merchants registered in South Africa. Similarly, if a processor is registered in the United States only for local businesses, then it may have to ignore all the applications coming from non-us territories.
To answer your question of how many merchant accounts exist, we can say that there are zillions as thousands of payment processing companies may have millions of accounts.
What is more essential for you to know is how many payment processing companies are offering merchant accounts in your country.
This can be easily identified by visiting the card brand website. Most of the card brand sites will have a list of authorised payment processors in specific countries. This way you will be able to know the names of all the PSPs in your country. After that, you can approach these processors individually and see which one offers the best services at the most economical prices.