The router for the 192.168.1.x network assigns each outbound connection a unique source port on the Internet side. When the router gets a reply, it looks up the port the packet was sent to in its NAT table and that tells it what IP and port to put as the destination on the LAN side.
When a packet is sent from the LAN side to the Internet side, the router checks its NAT table. If there's no entry corresponding to that local source IP address, local source port, remote IP address and remote port, one is created. A new destination port address is assigned. The packet is sent to the Internet with the source IP rewritten to the router's Internet IP and the source port rewritten to the assigned source port.
When a packet is received on the Internet side, the router checks the destination port in its NAT table. If there's no entry corresponding to that remote IP and destination port, the packet is dropped (unless something else the router is doing wants it). If it is, the destination IP address is changed to the correct local IP address, the destination port is changed to the original local source port, and the packet is sent to the LAN.
The "unique ID" you speak of is the combination of the remote IP address the local machine wants to communicate with and the local source port chosen by the router.