A mirror match is a match where both players are using the same character.
In this case, both you and your opponent need to be using the same ship AND weapon.
This essentially needs to be done against another player (local controller, dummy is fine), as joining with a second controller is necessary to select the opponent's ship and weapon.
This can be done in any game mode, although the recommended mode is tournament. This is because gladiator will return you to the main menu after each match, and championship is similar to tournament but less dependable.
To make each match very quick, you can have one of the sides suicide by pressing RB+RT+LB+LT. This will work towards other achievements, 50 suicides and 10 matches lasting less than 3 seconds.