Anonymity is not some sort of natural right. For most of our existence as a species, we lived in small groups where you quite naturally knew everyone you dealt with. True strangers were rare, and quite rightly regarded with a certain suspicion. Anonymity only became possible when we started living in groups large enough that you might have to deal with people you hadn't met before, because there were just too many people around for you to know all of them. And even in such circumstances, if you were going to enter into some sort of serious agreement, like buying on credit or renting property, you would absolutely have been required to identify yourself. Historically, anonymity of any sort has only sometimes been possible, and anonymity in serious matters has generally not been possible at all. It is therefore not reasonable to speak of a natural right to anonymity.
My position, strictly speaking, is that anonymity is generally permissible. If you want to try to remain anonymous, that is in many cases fine. But it is also quite difficult, particularly in the face of determined investigation, and is therefore rather unrealistic. Unless you really know what you are doing, your attempts will fail as soon as someone really cares about finding out. This makes combining anonymity with any sort of public prominence or celebrity status a particularly bad fit, because plenty of people care about knowing all sorts of details about celebrities, so there is plenty of reason for both amateur snoops and professional investigators to go looking.
I don't find your example of celebrity pseudonyms particularly convincing. These are simply terms of convenience, part of crafting a public image. They are not true attempts to hide anyone's identity. Pull up the wiki page of most any celebrity that goes by a stage name, and you'll find their real or original name.
https://en.wikipedia.org/wiki/Tupac_Shakur
https://en.wikipedia.org/wiki/Elton_John
https://en.wikipedia.org/wiki/John_Wayne