The link for the story in the UK Daily Mail about the Randi's remarks is (url) here.
Pseudonyms have long been used by actors, performers, musicians and authors, as was the case back in the 19th Century when female authors used men's names. In recent years, content creators have often invented nicknames for themselves and even trademarked them (sometimes to distinguish themselves from others); this could become a problem for any service that insists on a legal names policy (which appears to be the case with Facebook). Sometimes people have used pseudonyms to distance themselves (or "protect") their families or business connections.
Google-Plus appears to have a more flexible identity policy that encourages the use of a nickname if that is how someone is generally known; here is the link. This policy appears to be in a state of flux. Blogger does not appear to have such a policy from what I can see. (Google accounts and Blogger accounts have their own profiles and right now are maintained in two separate places; it wouldn't surprise me to see this change in the future.)
It seems to me that a service provider could allow pseudonyms and not allow anonymity or false or misleading names.
You can see a complete explanation of my "identity" for searching on my Blogger Profile.