It is not "owned" except in a weird metaphorical sense, and moreover, big banks are obligated to own an exact amount of shares - no more and no less, and they may not sell them - by act of Congress. If you scroll a little bit up from your link: "the 'ownership" of the Reserve Banks by the commercial banks is symbolic; they do not exercise the proprietary control associated with the concept of ownership nor share, beyond the statutory dividend, in Reserve Bank 'profits.'"

A "US Dollar" is a unit of measurement; a "Federal Reserve Note" is a physical object. You can no more own a US dollar than you can own an acre or a joule, but you can certainly own land or a battery. Casting doubt on the US dollar because you can't touch it makes as much sense as casting doubt on meters or Bitcoins or Goldman Sachs because you can't touch them either. They're all social constructs, but social constructs are quite real for anyone participating in the society that constructed them.