I think you are missing the key element, that socialism isnt just a structure or institution, but a government policy and applied to all people within a society

I think you are missing the key element, that socialism isnt just a structure or institution, but a government policy and applied to all people within a society. It is neither optional or excludable. This is why the monopoly man choosing to share caviar with his family or paying dues to a yacht club does not make them a socialist. The monopoly man retains full agency over who they share with.