It's a private company in name only. It was enacted through congress with a board of directors that are actually a federal government agency. Saying it's a private company is not even technically true.

It's gluing feathers to my iguana and calling it a bird.

Yes, but not really.

It's not surprising how many people aren't aware of that. When you look at the ceremony around the chair and reports to congress, when you read Wikipedia articles, it sounds as if we have a government owned central bank when in fact we have a private entity with distributed and opaque governance which is ostensibly accountable to the congress although I've not seen congress ever act like it has any accountability for fed behavior.