I think it's possible he can?

Neji doesn't have a lot of experience with affection, particularly physical affection. I admit that's sort of inferred, but considering that he spent years 4-14 pretty much hating everyone and definitely not seeking out physical comfort and affection, it seems pretty reasonable. What we see of the Hyuuga also tends to be pretty staid and not really demonstrative. When things are going well, we show it by... training together. And drinking tea. So it just becomes, like, a thing Neji isn't sure what to do with. Most of his physical contact in life is either professional or violent. He's fairly comfortable with physical bodies, but it's in a weird way. Gestures of physical affection, at this point, leave him feeling distinctly awkward. He's not sure how to respond to most of them, and there's this sense of them being sort of undirected? It that makes sense? Like they're not a very purposeful interaction. For Neji, there doesn't seem to be a goal. With hugging, it's only a little awkward. Cuddling would be more extremely awkward.

The exception to this, at this point, is sort of the non-human types of affection. I think that's sort of different, and it might sound weird, but there seems to be more of a point to petting Obsidian or Lark than hugging Inara. Not because he dislikes Inara—like I've said, he does like her, and he's generally comfortable with her—but animals are different, and that sort of demonstrably calms them down. And if they cuddle, well, that feels more like being a bed/pillow/whatever. It's probably something that will help him get comfortable with human touching though? :|a Like, a GATEWAY HUG. idk.

Also, Neji doesn't have anything in the romantic direction at all, which is cool and not complaining about that, but if he did, that would push him in the direction of getting more used to it. Sort of headcanon, since shounen manga! But in a way, I think he would feel less awkward kissing than just hugging or cuddling, because it does seem more directed to him, like there's more of a purpose to it, and it seems more active. So if that were a thing that came up, it would also probably acclimate him a little more to physical affection.

For what it's worth, Inara's done a good job clicking with how Neji does understand affection, like making him tea, bringing him water and a towel after training, and telling him to go to bed because he's overdoing things. I don't know if he'd identify it as such, but it's definitely there, and I think just as he gets more comfortable with that, he'll loosen up some more with her. Even if it might be easier to turn into a dragon. >.>

Maybe that makes sense. There were a lot of words.

Most important people for your characters?