Several of the socket options are handled at lower levels of the system. While most of the socket options could be set using the setsockopt. Reference:man setsockopt And since you are mentioning only POSIX on any Linux, in general, as your scope. The accept() (Reference: man accept) does have a certain amount of discretion on what socket options should be inherited and what options to reject from the listening fd.
accept() does not modify the original socket passed to it as argument. The new socket returned by accept() does not inherit file status flags such as O_NONBLOCK,O_ASYNC from the listening socket.
So, instead of relying on the inheritance or non-inheritance of the listening socket properties(which is bound to vary across implementations and licenses), the accepted socket should be explicitly set with the desired socket options.(Best practice)
man pages and the implementation codes in your machine would be the most relevant specification for the accept() behavior.There’s no common or standard specification existing across multiple variants of Linux.