Your DVD player is fine. The problem you are having is overscan. You can either re-encode your anime to fix this problem or reduce how much your TV overscans.
You should look at this thread for the first solution:
Convert AVI/MKV/MP4 to DVD .
If you own a relatively new HD TV, it might have a built in picture option called 1:1 mapping that should eliminate the overscan. For older tvs you can try to find the service menu and re-adjust the overscan. The side affect of re-adjusting the overscan is that you might see garbage on the edges of the screen when watching TV stations (The reason why TV cut off part of the picture).