Even after enabling client cache one still has to call SSL_set_session. See documentation of SSL_CTX_set_session_cache_mode point SSL_SESS_CACHE_CLIENT.

I started thinking about not exposing a SSL_SESSION object to the user but rather extending wrap_socket to take an already established socket as argument and use that socket's session object. This way I can ensure that both sockets share the same SSL context

I am not really convinced by this idea myself, what do you think about this? Any better ideas?