As in the character he portrays in his songs. Basically, is stuff like "Parents Don't Understand" canon to the show?

Parts of this show are based on Will Smith's real life. In real life, Will and Jazz were friends. In real life, Will was nicknamed by his teachers "Prince" because he was charming. If you saw the episode where Will and the Banks go to Philly, Will's old neighborhood friends call him "The Fresh Prince." So I think the "Fresh Prince" part of the show's title stems from Will Smith's own life story.