No, websites don't have to register with anybody. You only need to register if you want to have a nice domain name instead of an ugly IP address.
But you can download Apache now, set up a Web server on your laptop, and host a website. Google will even index it (if you have inbound links).
Does this mean you're not a "real" website? And when people talk about the growth of the Web, they should ignore your site?
No. In fact, in the early days of the Web -- back with there were only a few hundred sites up or so -- many could only be accessed via IP addresses. And, today, there are still many sites that can only be accessed via IP -- for example, if you set up the default Diva Distro, it automatically creates a website for you that can be reached that way.
You can also register a domain to point to that IP address to make it easier for people to remember your site's address, but that's voluntary.
You can have a site on the Web. You can have a site on your company intranet.
I see a parallel with OpenSim here: You can have a grid on the hypergrid. You can have a private grid that's behind the firewall, or that requires user logins.