You are never truly anonymous (unless you lie with your registration data - which has its own risk attached), but you can keep your data out of “whois” searches.
When you register a domain at many hosts, you may use their “proxy” info for the whois record. Some charge money for this service, others do not.
Dreamhost allows you to keep your info only on file at Dreamhost, and puts their “proxy” info into whois in its place, for no extra charge. You elect or decline to do this upon registering the domain at panel.dreamhost.com, domains, register.