To clarify for readers: Vereenigde Oost-Indische Compagnie aka "Dutch East India Company" actually properly translates: "United East India Company" ("Dutch" helps distinguish from "British").
Despite the merchant face (or candy coating?), the DEITC(VOC) and the BEIC are probably best seen as military expeditionary forces of their respective People/Crown. Perhaps the historians and propagandists like to spin them up as "merchant companies" so that the respective Crowns don't get blame for any stink of tyranny or cruelty. So it could be said that when the British took the United Netherlands, they took over the DEITC military company. The Treaty of Paris is whereby the British Crown yielded sovereignty over the BEIC and any other expeditionary companies n America. The replacement of the Union Jack with the stars of the sovereign states of America simply reflected the change.
The imperialist behavior of the USA after, say, 1800 may simply have reflected that they were very much aware of control over local BEIC/DEITC operations and so, they 'tooled' around to say the least.
The Battle of Manilla Bay.
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To more succinctly address shikamaru, consider Black's Law Dictionary definition of state. The British East India Company (actually named "Governor and Company of Merchants of London, Trading into the East-Indies") was by definition per its charter "one body corporate and politick" and a "fellowship" to boot.
The "body politick" part was at least extant. The US Army Rules of Land Warfare suggest that the occupying military is not sovereign. Makes sense. BEIC and DEITC/VOC weren't sovereign either. They made acquisitions for their respective Crown or People."state" OR "independent state"(?): a people permanently occupying a fixed territory bound together by common law habits and custom into one body politic exercising, through the medium of an organized government, independent sovereignty and control over all persons and things within its boundaries, capable of making war and peace and of entering into international relations with other communities around the globe.
I would not go as far as to suggest them to have been private armies. BEIC was chartered directly by Queen Elizabeth I. I'd tend to see it as very much the British Navy bundled with a merchant operation. Can you say: admiralty?