Whenever someone gets a final kill I always see their player card and emblem on the kill cam, but when I make a final kill it only shows the gun and not my player card. How do I get it shown?
This is expected value.
The killcam is a bit of a funny thing, because it is not a video recording, but rather a replaying of events. In previous CoD games (like MW2) it would sometimes even give you a second credit for these kills as a result.
The reason you are experiencing this behavior is because when you are following another player it will show you their player card, however, the game does not do this when its following you (as then you'd see your player card constantly as you played). Because the killcam is a replaying of events, and because you got the last kill, it will show you exactly the same thing as when it happened.