Exactly. They have done a great job with branding to make almost everyone think they are an official part of the government. The Chamber of Commerce represents generally support incumbent business interests.
It seems like the "of" does it more than anything. It parallels organizations like "Department of Defense" and "Federal Bureau of Investigation": some generic noun to represent that it's an organization of people, "of," and then an abstract noun that gives the organization a purview that's universal or total in scope. Most organizations that have the audacity to do this have a monopoly on the use of force behind them.
The CoC, of course, is itself in the business of government: trade its governmental access and leverage to businesses in exchange for large cash grants.
It's even better than blurring the perception between private and official bodies. They now run an "accreditation" program for local chambers of commerce, some of which have membership that would hold divergent views. This national organization is a lobby for big business and has little regard for small town business owners.
However, I don't think it's the fault of "the." Nobody thinks The New York Times or The Washington Post are government entities, for example.