Was there an equivalent to The East India Company in the French and Spanish colonial empires; a company that came to be regarded as a national asset and effectively ruled a particular colony?
Yes: for the British Empire alone you have the Hudson's Bay Company, for one. For the Dutch there was their own East India Company - the VOC, and for the French, there was the Compagnie des Indes Orientales, the latter of which competed directly against the British EIC in India, especially in the South. Each of these also had West Indian equivalents. For Spain, I am not sure, so I will leave it to someone else.