The WIC, like its counterpart the Dutch East India Company (or VOC) and the English East India Company, were peculiar hybrids whose emergence characterised what can be considered a new stage in the economic history of their countries. It's this that produced the unusual results that you're referring to.

Essentially there are two key considerations to bear in mind here. The first is that the various Companies competing in both the Caribbean and the East Indies were targeting markets for resources that were in high demand in Europe, but could not be easily produced there – sugar in the case of the WIC, and spice in the case of the VOC and its English equivalent. This made them potentially vastly profitable and attracted large investments. These investments could be (and were) further leveraged by using new instruments introduced precisely to service the new international markets, which we're still familiar with today – the stock market and futures trading. The result was that all three companies were sufficiently well capitalised to be able to afford to take what they wanted from both the indigenous peoples of the regions, rival empires such as those of the Spanish and the Dutch, and each other. They did this by recruiting and outfitting their own armed forces – naval and military. It was these private forces that did the conquering and seizing that you refer to, and the lands and the resources taken were ruled in the name of the company that took them, not the nation that the company was based in. The VOC, for example, very explicitly made it clear to the rulers of the Dutch Republic that its new possessions in the East Indies were not, and should not be considered to be, state possessions.

However – point 2. All these companies were operating in a politically unstable world, and the best way of maximising profit while minimising risk was definitely to have the backing of their respective states. This support came in two forms: firstly, the state permitted the companies to establish bases and to recruit and outfit units and build and repair and equip ships in its territories, and secondly the states agreed with the companies that they would enjoy a monopoly on the importation of their luxury goods into its territories. These concessions very significantly increased the value of the businesses concerned, and naturally the states wanted something in return. That "something" comprised customs duties on imports plus a very substantial cash payment to the state, which was offered by the companies in exchange for a formal charter outlining and confirming their agreement and setting out their privileges. All three of the companies we're considering here, then, were "chartered companies", as were their numerous foreign rivals (there were Danish, Swedish and French Indies companies as well). The charters that were issued also had to be regularly renewed, offering the states the chance to renegotiate the terms of their agreement and the sums paid for the charter.

In consequence, although all the major India companies were privately owned commercial operations, they were also significant landholders in the territories they operated in, and needed to establish armed bases from which to trade and maintain armed forces to defend them. On top of all that they can be considered, effectively, to quite a large extent valuable state enterprises – to the extent that the English East India Company ended up as de facto and de jure ruler of India, which did not pass formally under British government control until 1858. This explains the need for governors and the tendency to require that settlers swear allegiance to parent states that you mention in your question.