The White privilege came from her not losing any status when caught, unless you believe that was due to some other factor like gender, economic status, or the like. But since the Progressive Left, where she is politically, and who are the leading proponents of the existence of White privilege, did nothing to her as a White for lying about her ethnicity--normally a politically and socially fatal thing--it argues that they gave her a pass based at least in part on her actual ethnicity.
And it's Different when a DEMOCRAT does it.