Audiobellum this depends on the game and has been changing each generation.
back in the day, quake 3 implemented the first first person/third person mechanic. they managed this by splitting the characters in 3, and needed it because it was a multiplayer game.
the body is split in 3: the legs, the torso, and the head. each rotates independently.
and then, most games of the time also had a third person and first person models. the first person model was just the arms and was rendered in front of the camera, while the third person model was either a single mesh or split into 3, and used during reflections and in multiplayer.
each required a different and unique set of animations to work, animations that you have to do yourself, you can't use the ones from mixamo, etc.
as technology progressed, first person games started to use whole models, like mirrors edge. in this game there was a special set of first person animations, and third person used another set.
first person animations look awkward from third person, but look right in first person, they are usually more stiff, as they fit to the limits of the camera.
and then you have the unity tutorials. don't follow those, they lead to clipping and the player can see their own face if the camera moves wrong. some games used this, it's awkward and looks wrong, but it's the cheapest easiest way. just a third person model and the camera attached in front of the face.