"Opheliac" by Emilie Autumn uses the psychological version of the trope. "She speaks in third person/So that she can forget that she's me."
Tim Minchin's Rock And Roll Nerd doesn't want to seem self-obsessed, so he writes his songs about himself in third person.
Soulja Boy Tell 'Em usually starts with his songs with "Soulja Boy Tell 'Em". (Nach.)
This doesn't happen all the time, and it doesn't seem to happen as often as it once did, but it has been a recurring feature in rap/hip hop music for the lyrics to include the artist or group referring to themselves in third person. This often happened especially during the 1980s and 90s.