Harley (Nice), Bolan (Feld), Bowie (Jones), Sylvian (Batt), Jansen (Batt), Numan (Webb), Ant (Goddard), Starr (Starkey), et al. Most artists don't choose this route.
Perhaps this was just a fashion thing but perhaps not, because this hasn't gone away (particularly, musically, in the rap genre) and some of course, just use one name, whether (a) birth name or a performance (created) name.
Members of Split Enz, in contrast, kept their names, but most band members used their middle names in preference to their first names, and retained their surnames (also) as their performing names.
A matter of choice and individual personality, including the workings of the individual ego, I guess. Language, influence, culture and identity, must play some part. Sitting Bull, Red Cloud, Geronimo, all spring to mind. Perhaps a desire to be...Perhaps (today) even something as mundane as accounting and or legal clarity, a separation from the private person's domestic activity.
Whatever, a system that is unique to homo sapiens, the human menagerie, as far as is known...and more than one name only became common post the Norman Conquest, after 1066, with the SLOW development of land law (the really modern system with which we are all familiar, is barely a century old, circa 1925). Widespread property ownership - land, and eventually, intellectual property, which is really what (we) are talking about here, is still really, not that old in the context of about 200,000 years of homo sapiens history (until recently it was thought that we've 'been around' as a species about half that of the new possibility).
I didn't visit London before 1972. Only in the mid 1990's, did I get to know it much better. When I lived in the North of England, about that (1972) time, one address that was familiar to me was 'HARLEY STREET', famous for the 'preservation' (health) of the individual, or even, to some extent, the 'remodelling' of the individual. I think that Steve chose well. Cockney Rebel's original delivery, was pure, confident, LONDON...