It is a GSE (government sponsored entity) created by FDR. That is why it is socialism. If FDIC fails the taxpayers bail it out.
BTW, the FDIC is not a GSE, like FNMA or Freddie Mac. It is an independent agency of the Federal Government. Yes, it does have the ability to borrow from the US Treasury.
The only way for the FDIC to fail is for its member banks to fail en masse, and wipe out the assets in the Deposit Insurance Fund. The FDIC has done a marvelous job of cleaning up the banking system.
OTOH, your industry (Wall Street) failed, and the taxpayers bailed you out. I haven't seen any tears of shame coming from the titans of Wall Street.