In an FPS, when a character runs, from the view of another character, the gun moves, and similarly when the character aims down sight. Are the views from the third person and from the first person actually showing the same gun position? I feel like it would be tricky to that happen and have it look natural from both perspectives.
The gun position is not the same when viewed from the FPS player perspective than from the 3rd other players perspective.
This is due to the fact that both of these views need to look "realistic". And there is no magic potion to make the look and feel applicable for both of them.
In the first-person-view, you rarely see the character's body, so what the player sees is only the hands and the gun, tuned so that it has a look and feel realistic.
And the 3d artists can work with the models that you see in 3rd person so that they too look realistic.