Andrew, I am delighted that you are ‘enthralled by James’ manner of explication and want to learn a bit more’. As you progress with the course, I am sure that you will find it truely life changing.
James’ use of the phrase ‘who you are is something which the social other produces in you’ means I think, just that. He is referring to the process of inculturation, that is, our instinctual practice of ‘desiring according to the desire of another.’ which begins in infancy with the influence of the parents and siblings, and continues with our schooling, and later social contacts and workplace etc.etc. In fact, for the rest of our lives. Let’s take for example, the very obvious question of language. As James tells us: “It’s not merely that we learn words from imitating the sounds of people who are other than us, we find ourselves being inserted into a language. The language was here before us and will continue after us.” In your case I assume the language was English. You were inculturated into this language by all those around you, that is, the social other. A child born in Beijing will not being to speak German, but will imitate the sounds he/she hears from those close. This is the case of every aspect of our lives, the social other is a model for us.
As to the question of the social other being prior to us. Consider just the one example of language, we learn this as if we were stepping onto a moving train. The train existed before us and will go on after us, the language existed before us and will continue after us.