Avi K, the reason Jews used such names (Norman, Morris, Arnold, etc.) is that they sounded sophisticated and upper-class to people at that time. It probably helped with upward mobility (and assimilation as well).
To answer Lightbright, I don’t know any gerim or born Jews who have done that, and I wouldn’t think it’s a good idea (for the identity-related reasons you mention), though I would think it’s forbidden or a horrible thing to do.
Nowadays, when there is no little anti-Semitism compared to the past, and so much assimilation, Jews should give their children Jewish names, instead of having two names, as people did in the past.