For internal communication on IPv4, everything has some unique internal IP. There are blocks reserved for private space. Usually people use 192.168.x.x or 10.x.x.x. DHCP hands it the address.

If you wanted this to work in the IPv6 world, you are assigned a prefix by your ISP, and everything is inside that prefix. Services still have to discover each other by some mechanism. Perhaps by DHCPv6, or perhaps broadcasting their existence.

Port forwarding is only necessary with NAT. If you have a gateway firewall that blocks incoming new connections by default, then you will need to open the port going to a specific device. Current home networking "routers" combine port forwarding and opening the firewall together as a convenience, but there's no reason an IPv6 world would need to do that. UPnP can open the port the same way if you want that (though that's a whole other security issue).

In a home networking "router", the gateway firewall is already combined in. In fact, I'm putting the "router" in quotes because it's really a firewall with NAT and some other services like DHCP. It doesn't typically do things like BGP that we would normally see in a router outside of an edge network like your home. A router out there is an allow-by-default device.

Adding NAT to the gateway firewall makes the code more complicated. For example, here's a command on Linux that activates NAT for the iptables firewall:

iptables -t nat -A POSTROUTING -o eth1 -j MASQUERADE

That "MASQUERADE" bit is handled as NAT, and iptables has to implement more code just to do that.

If we wanted to simply drop all new incoming connections, we would do:

iptables -P INPUT DROP
iptables -A INPUT -m state --state ESTABLISHED,RELATED -j ACCEPT

Which tells it to drop packets by default that aren't otherwise accepted, and then accept packets that are already part of a connection. Even with NAT, we typically want to do this, anyway, so we're not making things any easier with NAT.

If we want to add a service listening on port 80 for host 10.0.0.5, we would do:

iptables -A INPUT -p tcp -d 10.0.0.5 --dport 80 -j ACCEPT

Which works just fine in a NAT-less world. With NAT, we also have to add this:

iptables -t nat -A PREROUTING -i eth0 -p tcp --dport 80 -j DNAT --to-destination 10.0.0.5
iptables -t nat -A POSTROUTING -o eth1 -p tcp --dport 80 -d 10.0.0.1 -j SNAT --to-source 10.0.0.5

Which translates the stuff coming in from outside to port 80 to 10.0.0.5 on the same port, and then also translates replies going back the other way. And I might be getting some of the commands wrong, because it's been a while since I've had to configure this.

Suffice it to say, dropping NAT greatly simplifies firewall rules. Your home router is still doing all this (many of them are just Linux iptables these days), but it's hiding the details from you.

Edit: This doesn't cover how protocols have been designed to work around NAT, and has resulted in a more centralized Internet that's easier to spy on. That's a whole other problem that is hidden from most people.