Well...
You're on Internet via your provider who provided you an IP-address. Over this IP-address all your traffic is going. Even if you're using NAT, the traffic finally goes over the IP your provider gave you. RIAA is collecting IP-addresses and asks your provider which name/address belongs to xx.xx.xx.xx, which they can find in the logs. So next they are going to that address... and there are you... behind a possible NAT which is in the same house where you're playing on a PC with 192.168.x.x (after the NAT)

All the stuff about anonymous... I'm sorry, but on Internet this is almost not possible.

Eize