There is some merit to the idea. As it is, you're already an independent actor from your monarch, which certainly corresponds far more to colonial companies than direct royal colonies.
What's missing from the concept is the need to pay off your investors.
Imagine you start out with a colonial charter from the monarch and some amount of gold (which varies by difficulty level). But you can sell shares (say in 1/20th increments) to raise money to buy ships, arms, tools, trade goods, recruit/train colonists, etc. But the higher the investor share in your company, the more you have to pay them off when you sell things (just like taxes) or otherwise run into gold.
The concept would have to be limited enough not to detract from the game, but significant enough that it lends some additional realism in terms of the constraints encountered by early colonial entrepreneurs.