This is a great question and deserves an answer. It’s a little outside my area of expertise but maybe I can at least provide a place to start. Spain’s international commerce endeavors began nearly a century before the British East India Company was established and thus operated in a completely different economic climate. Initially, the Crown was very concerned about instituting direct control over the Indies and famously maintained a strong monopoly over New World trade for much of the colonial period. As a result, very few private companies existed as exploration, conquest, and trade was controlled by the Spanish Crown.

But during the eighteenth century, it became increasingly clear that Spanish policies would need to modernize to keep up with the British, French, and Dutch. As part of the Bourbon Reforms, Spain chartered several companies modeled after many of the famous companies that you mentioned in your question. The most famous of these is the Royal Company of Guipúzcoa (Compañía de Guipúzcoa de Caracas) which controlled the production of tobacco and cacao. It was run by Basque merchants to maintain a monopoly over trade in Venezuela and quickly gained control over the local economy, proving wildly unpopular among local planters who rapidly fell into debt. Another was the Royal Company of the Philippines, chartered to maintain a monopoly of Asiatic trade. The company was chartered several times but never managed to remain profitable for long. In both cases however, the Spanish Crown only allowed private companies to operate in the peripheries, which greatly reduced potential investment and profit. They also never seriously challenged the royal convoy system. As J.H. Elliott explains in his book Empires of the Atlantic World, “the companies were only allowed to trade with marginal regions of America, like Venezuela, which were not directly supplied by the transatlantic convoys, the Andalusian monopoly--considered essential for the retention of control over the silver remittances--remained largely intact” (232).

It is also worth noting that from 1581 to 1640, Portugal and Spain were united in the Iberian Union. Thus, the Crown maintained ultimate jurisdiction over Portuguese companies. However, these were little affected by Spanish control and continued to primarily benefit Portugal rather than Spanish investors.

Finally, we also must consider the contingency of independence. Prior to the revolutions, many in the Spanish Empire expected drastic change to colonial economic policies. The Bourbon monarchy had already instituted modest free trade reform, with more expected. Since events that led to independence were so spontaneous, there is no telling how these reforms might have played out or if they would have led to the rise of other private companies; that, however, exists outside of our task as historians. I hope this answer provides you some direction!

Sources:

• Empires of the Atlantic World by J.H. Elliott
• Silver, Trade and War: Spain in the Making of Early Modern Europe by Stanley Stein and Barbara Stein