If you put someone else's game into your gameboy the saved game will be the other person's because the game is saved onto the game card, not the game system itself.

Yes, it can. It corrupts your saved game, and sometimes when you turn your game back on, it doesn't even go to your last saved game.

If you want to start a new game, it's right below the saved game of yours.

WARNING: SAVING A NEW GAME WHILE YOU ALREADY HAVE A SAVED GAME WILL DELETE THE SAVED GAME.

all of the data for a certain game is saved on the game card for it

No. It shows a warning that if you click new game you old game will NOT be saved.