When you use the word "countries" I feel that it is safe to presume that you mean nations- Technically, North America has more nations than South America, as continental NA does not end until the tail end of Panama. Furthermore Greenland is a part of NA as well- thus NA is one of the largest regions of nations stemming from a rich history and a high degree of cultural diversity unparalleled by much of the world due to universalist and libertarian values. Such values also play a major role in North America's mixed economic activities.
Furthermore, the states that you are speaking of are smaller countries that became members of federations many years ago- such as the United States of America and the United States of Mexico. Thus North America not only has more states and nations than SA, but also has the most countries (as in cultural and land regions). Moreover, the day will come when the UN federalizes and each current member nation-state will be its own country (as seen in the USA and USM), through membership to a global federalist power known as the Federation of United Nations. This presumably has already begun and will continue throughout each generation via the stable unification of culture, language, economies, etcetera.
Hopefully this answer leaves new doors open for you to explore more questions regarding the unification of countries as nations and nations as federations or unions.