The stock of member banks in the Federal Reserve cannot be sold, transferred, or pledged. The stock is actually in the nature of forced interest bearing loans from banks. Some nonprofit organizations like apartment co-ops and private clubs have a similar structure, with members required to put money in when they join and getting it back when they leave. In both cases, part of the reason for such a system is to weed out those who are unable to spare significant money to become and remain a member.
Congress could abolish the Federal Reserve member stock system. Doing though so would require paying the member banks back for what they had put in and would tend to open Federal Reserve membership to weaker and less substantial banks that had an interest in diminishing the regulatory oversight of banks and other financial intermediaries.
The creation of the Federal Reserve System was hotly debated at the time. I do not know what the vote in Congress was.
The following two papers provide useful accounts of the origins of central banking in the US and of the formation of the Federal Reserve System:
Historical Beginnings… The Federal Reserve
New York and the Politics of Central Banks, 1781 to the Federal Reserve Act
It is not. It is privately owned.