It's pretty common for musicians to use nicknames or stage names instead of their real names; for example, Flea from Red Hot Chili Peppers and Bono from U2.
Is there a reason why so many musicians do this?
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Some of them (noted in your question) are nicknames that they had before they were famous. Other times, names are changed by record execs to sound catchy. John Mellencamp was renamed John Cougar because an exec thought it sounded edgier.
Other times, it's just a matter of privacy. Musicians have been known to make up stage names so they could still have a number listed in the phonebook and not get their house swamped by fans. There's also a little mystery about not having your background traceable. You can make up any backstory you want. This isn't as easy to do nowadays due to how easy it is to fact-check in the "Information Age".
There is a list of reasons with great examples in this wikipedia article.
Various reasons include avoiding being linked to an existing celebrity (whether it's an actual family relation or not), easier pronunciations, to avoid discrimination, building a brand, or just something catchier.
One reason is that in some genres it's the thing to do. In both black metal and hip hop, it is rare to find artists that do not use stage names. This is likely due to that the originators of the genres started using stage names for different reasons, one of them usually being that the stage name sounds cooler than the real names.
Another take on this: some people feel that they're different on stage, and a stage name just encapsulates that feeling. For example, Hank Williams' "Luke the Drifter" alias for his religious-themed material. An example from another sphere, UK standup comedian Jenny Eclair ("Jenny" is real name, but "Eclair" is a stage name) often refers in interviews to things that "Stage Jenny" will do that the other Jenny might not.