This is seemingly possible, and preferable in the scenario of an untrusted network. Some (rare?) DHCP servers may however require this field (as an administrative choice).
In my crowded, trusted home network, the DHCP server unfortunately only displays this field in the listing of current leases, so it is a pain when a client does not send a client-identifier (or sends a relatively meaningless client-identifier).
For Linux computers it may be possible to configure the DHCP client to remove any connection between the hostname and the client-identifier used for DHCP. As an example of where this might be needed:
The computer that I am currently sitting on has a wired interface and a wireless interface. Both interfaces are using DHCP. They show up in the DHCP lease table with the same name (i.e. nothing to distinguish which interface is which), that name being the hostname, presumably because Linux sent the hostname as the client-identifier in both DHCP requests.