[Hikikomori]

IP has no ports, it has a protocol field, TCP, UDP, IPsec and other layer 4 protocols have their own protocol number. NAT (PAT really) uses TCP and UDP ports to connect what can be considered separate connections, ie a connection on the inside and a connection on the outside, these connections don't even have the same source/destination ports as connection state tracking is what connects these separate connections.

The socket API has different types of sockets, typically you use stream (TCP) or dgram (UDP), but you can also get a completely raw socket where you need to construct your own ethernet and IP headers, or create your own replacement of IP protocol. So socket does not mean only TCP/UDP or something with ports, unix sockets are another example of this.