The East India Company was a real-world English (and later British) joint-stock company, founded by John Watts and George White. It traded in the Indian Ocean region between December 1600 and June 1874.
In the game, the company is referred to as HEIC and the Honourable East India Company. In 1807, it sent the chief inspector of its London office to investigate the mysterious reappearance of their vessel Obra Dinn, which was missing for many years.
For more information, visit the article about the company on Wikipedia.