I read that NICs in modern routers are responsible handling of multiple ports.
A router port is normally equivalent to a host NIC. Some routers support switch groups where the layer-3 'port' is more a virtual NIC or a router port connected internally to a built-in hardware switch.
Is it true to say that every port has a different MAC and IP address, and thus a NIC might have multiple different MAC and IP addresses (corresponding to the appropriate ports)?
Normally, yes. A router requires multiple ports in order to connect to different subnets. Each port (not necessarily equivalent to physical ports, see above) uses its own MAC address and IP address and each port may have multiple IP addresses (also pretty common with hosts) and even MAC addresses (e.g. virtual MAC addresses in failover scenarios).