First, not all of Latin American colonies were Spanish, so obviously Portuguese, French, and British holdings developed somewhat distinctly and their differences in power structures led to natural divisions upon independence. As for the Spanish colonies, during Spanish colonialism, the Spanish set up administrative divisions for their vast holdings in the Americas in the 1500s. They established two very large geographical divisions called viceroys, with one based in Mexico City that controlled all holdings of North and Central America and another based in Lima that controlled all holdings in South America. Those viceroys in turn established smaller units called audiencias, which in Spain was a regional court, but in the Americas also had some autonomy and certain legislative and administrative authority. The territories of these audiencias were geographic divisions of territory that, after three centuries of colonialism, resulted in natural breaking lines when the relatively fast independence movement in Latin America occurred in the early 1800s. The first audiencia was located in Santo Domingo in 1511, and it oversaw the Caribbean Islands and nearby mainland. Over the course of three centuries, the Spanish wound up establishing 12 audiencias, most of them based in the capital cities of the current Latin American countries. In the late 1700s, the Bourbon reforms caused dissatisfaction from many of the Creole who had power in the audiencias. Then, during the Peninsular War in Europe, the Spanish crown supported the French against the British, which led to the British and their superior navy severely cutting off Spain's contact with it's American colonies. That lack of contact led to further concentration of power in the audiencias. Then the Spanish crown influence was seriously in question in the 1800s when Napoleon turned on Spain and put both Ferdinand and Charles (the king and former king of Spain) in prison. Rather than the very large and cumbersome viceroys uniting the Americas, the natural breaking points at that time were the audiencias (and some military divisions as well). By the time the various independence movements occurred through Latin America, there was far more unity around the audiencias than the viceroys, so the new nations split accordingly, with those lines shifting later after various wars.

As the other redditor has pointed out, the division of the hispanic america into different countries was mostly related to "Audiencias", according to most historians. The "audiencias" were political institutions formed by "oidores", which had to rule a viceroyalty or a governance (like Chile or Venezuela). They had more power than the viceroys, the other big power at colonial times, like the capacity to exercise justice, and both disagreed very often. The audiencias also grouped up the elites of each viceroyalty or governance and taked decisions on how to govern the territory.

By the end of XVIII century, there were an "audiencia" in Santiago, Buenos Aires, Lima and the others latin american capitals. Each one of them supported the creations of different countries through Latin America during the independence's movements. It has been discussed a lot, but most historians agreed that elites preferred to keep their powers, instead of creating a new "big" country by putting together different "audiencias", all of this in a context of social chaos around the independence wars. To see how important were the audiencias to the independence's process, countries in Latin America defined their territories according to the notion of "utis possidetis", which means they had to keep the same territory that "audiencias" were ruling by the end of the colonial period. So, Argentina was founded inside the limits of Buenos Aires' audiencia, Chile over the audiencia of Santiago, etc.. Only the audiencias of Cusco and Guadalajara, the newest and weakest ones, were absorbed by Lima and Mexico city.

After that, there were different political projects to unify the countries, but all of them failed very soon and the new countries decided to follow their own paths. Even today, there are many latinamericans claiming for the integration of the "Patria Grande" ( something like the big homeland), but it is another story.

P.D: Sorry for mistakes in the use of english.

Also there is the influence of the criollos (the white elite born in the Americas). Their goal was to trade and relate with Europe, and their identity was much more close to Spain than with each other.