How do shared IP addresses work and do they give privacy?

I assume "shared IP addresses" work by NAT, because obviously two devices with the same IP address need someway to distinguish them.

I've seen some privacy advocates boast they have "shared IP addresses" making it impossible for someone to track their activity.

For example

IPVanish takes an interesting approach to privacy and security. They use shared IP addresses, so when they say no one has any idea what you're doing when you're connected, they mean it.

Is all that is happening that connections on a specific port go to a certain computer and in that sense, while knowing the IP address isn't enough, knowing the port and IP would uniquely identify a computer? And IP packets contain the source/destination port so this really isn't saying much.