Could be. But the East India Company was sort of a monopoly, sort of a semi-governmental enterprise. I don't know the history very well, but it was pretty easy for the running of India to be transferred from the company to a British bureaucracy that took a dim view of commercialism.
Maybe it would be better to say that the empire may have liked big commercial ventures, large trading companies, and plantations, but didn't want a lot of small English businessmen around making India too much like the England they'd left.
Star fleet might be a private enterprise, but it wouldn't be what we have now, at least in science fiction, because sci fi values the emptiness and differentness, the strangeness of space too much. If it's too much like earth, the escapism is gone.
And while the space of sci fi films (more so than the novels) is dominated by government or quasi-governmental entities, you probably won't see Spock and Kirk or Han and Chewy filling out tax forms, since people don't want to be reminded about that either.
I am going to chew on that. I am in the process of coming up with a story...