During the fight between Neji and Naruto, Naruto comes out of earth to finish his final blow on Neji. So does he possess earth element as his chakra nature?
No, in that particular fight, he just (dug a hole and) hid underground to surprise Neji. No part of that used Earth chakra or Earth jutsu. I don't think he even knew any Earth jutsu. He eventually got access to all chakra types due to the Six Paths mode, but still, he was never seen doing any specific Earth jutsu.
Only thing i would add is upon review of this scene, they make a point to show Naruto stomping a hole in the ground as he's running towards Neji for the last time. He shouldn't be able to do this, as the other person who uses this trick (Kakashi) uses Earth chakra to do so. Think this may be an instance of plot armor unfortunately.
The tunneling scene was not an example of earth style, however, the question itself shows a misunderstanding of nature manipulation in general. People don't "have" a chakra nature. People have an easier time learning to change chakra to specific natures.
Most people have one specific elemental nature that they can grasp very quickly (relative to other natures), and usually have at least one more element that they can learn faster than the rest. There is nothing preventing somebody from learning to manipulate chakra to every single type of nature, other than time and talent.
For example, if Naruto had wanted to, he could have learned lightning, fire, or any other elemental manipulation when he first learned of the concept. Wind was simply the most practical for him to learn, as it was the nature he is innately suited to use.
No,because when he fight with kaguya he use all it chakra of tailed beast(1-9) he used sage art:cho biju rasen shuriken(sage art:giant super tailed beast rasen shuriken).so he used up all biju chakra(magnet,blue flame,water,magma/lava,(im not sure about kokuo maybe fire and water),acid,(i dont know about chomei),ink,wind)so is not revealed naruto using earth chakra/earth style jutsu.