A TCP session is a unique combination of client IP, client port, server IP, and server port.
So you can use the same IP and port as long as the destination is a different IP or port.
This means that in principle you could use the same IP and port to connect to every IP address on the Internet using 65536 concurrent sessions. 😆
This wouldn't help going to popular destinations, since they have a lot of people going to the same IP address and port, but for many (most?) of them you probably have some sort of CDN servers in your data centers anyway.
Nixon was a potty mouth.