> Remember, your identity in a TCP connection is your IP address
The essential tuple of a stateful IP session, the "5-tuple," is the source IP address, source port number, destination IP address, destination port number and the IP protocol number, in this case TCP. None of those elements can change during the lifetime of the connection. Or rather, changing any element constitutes a different connection.
Not that you needed to be told that; you probably know this already. It's just an interesting factoid that I imagine others might find interesting or useful when thinking about networking. In the guts of every device that handles TCP state (clients, servers, middle boxes, whatever) there is a table of all in-flight 5-tuples that marries every incoming packet with some state.