I've been managing domains since the mid-1990's, before Google or GoDaddy were even an idea. I have no idea what the internal policies of those two companies may be, but speaking in terms of law and the adapted policies of most hosts, you can create an anonymous registration for most business domains (e.g. ".com" domains), but not for personal domains.
Just to be sure we're all using the same terminology:
domain: You lease that.
www.yourname.org, or whatever it is.
registrar: The companies that facilitate domain registrations. They say "bcs001 owns
www.yourname.org, and that domain name points to xxx.xxx.xxx.xxx IP address."
WHOIS: This is the official and open list of domain registrations. You can look up your domain, and see how it is registered.
host: The IP address to which your domain name points, and usually where your content resides. This may be the same as your registrar, or a completely different company.
My registrar and hosts have mostly resided on two separate continents, but I suspect those using GoDaddy or Google as their host, probably just also register their domain name thru these companies. Who the actual registrar is might be a third party, or not.
Depending on domain type (e.g. ".com" vs. ".org" or ".us" or ".fam"), anonymous registration may be allowed. Check with your registrar. In this case, the registrar becomes the name and address of record, and only they know how to directly reach you.