The Arduino with Ethernet shield doesn't HAVE a name. It has an IP address. DNS is the service that resolves a name to an IP address. The Arduino doesn't do DNS. If it is a client, it can make GET requests using names, IF there is a DNS server available for it to use to resolve the name to an IP address.
If it is a server, some other process must be involved that calls DNS to resolve the name you use to an IP address. But that process won't involve the Arduino.