What does “born on the cusp” mean?
The phrase usually refers to a birthday close to the popular boundary between two Sun signs. Those calendar dates are convenient approximations, not a rule that gives someone two Sun signs.
Why the boundary moves
The Sun enters zodiac signs at a specific astronomical time, which varies somewhat by year and time zone. Someone born near a boundary should calculate the exact solar longitude for the birth moment.
Why people may identify with both signs
A neighboring sign can still be prominent elsewhere in a chart, and Mercury or Venus may be nearby. But this is different from having two natal Sun signs.