So. The short answer is that I don’t know. I’m not sure anyone does. But it’s my belief that the CIA is known as ‘The Company’ as an allusion to historical entities known as ‘Companies’ - and most specifically of these, the East India Company.
For those that don’t know, the East India Company was formed by Queen Elizabeth I in 1600CE, ostensibly to manage trade with the Indies, for the then-fledgling British Empire. Over the next 250 years, the EIC not only managed trade, but it wielded unprecedented power over a variety of countries in the region - sometimes directly, but more often through puppet rulers that the EIC had chosen and placed itself. It controlled and armed sympathetic local warlords, and even developed its own surprisingly strong Army and Navy.
So - I wonder if the CIA are known as ‘the Company’ because of their perceived interference and control of a variety of countries - in the Middle East, South America, and the Far East - as an allusion to the EIC of British history, and other similar ‘companies’ from the time. Happy to be proved wrong by anyone who knows better!