> For example, in IPv4 each host has one local net address, and the gateway uses NAT to let it speak with the Internet. Simple and clean.

I assume you mean "interface", not "host". Because it's absolutely not true that a host can only have one "local net address".

EDIT: a brief Google also confirms that a single interface isn't restricted to one address either: sudo ip address add <ip-address>/<prefix-length> dev <interface>