from a technical standpoint, there is several layers to this

ARP is how the routers talk at the base layer

IP is a secondary layer that provides routing capability

having a private IP on a non-routeable address range can still make outbound connections through a router that does have a public routable internet IP address.

how it looks internally, if you can connect a PC directly to the outbound-only capable network is your ETH device shows it has no address, but it has a MAC address which is how ARP from the router's PHY can find the path back to send you the responses to your requests

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i also should point out that a lot of cheap domestic home wifi routers have "wifi isolation" on by default which means you can't make connections between your devices both connected to the same wifi, without the additional use of a VPN to do rendezvous of your connection for you