Giga Wing
Dreamcast

Play as Stranger:

Go to the Gallery section and press B, X, Y, B, B, Y, X, B at the main screen. You should here a sound to confirm.
Another way is to beat the game under any difficulty setting without using continues.

Player Two Gets Invincibility:

Choose the Two Player mode, and have Player One pick a jet. Then, press Start + A on the second controller when picking a jet for player two.