Hi Ekaterina. No, it would be more appropriate to say that the person is "in" the seat next to you. Even though he or she is literally "on" the seat, the proper way to describe it, idiomatically, is "in." Also, I wouldn't refer to that person as your neighbor, unless he or she lives next to you or on your street.

I'm not completely sure this answers what you are asking, but from what I understand: If you are trying to refer to someone next to you (i.e. in a theater) you might say: The person next to me is eating popcorn. The person on my right has candy. The person in the neighboring seat is talking loudly. In the example you proposed, I would almost always say that you'd need to be more descriptive about the person. If you just say something like: My neighbor seems to love candy. I would assume that the person who lives next door to you enjoys candy. My neighbor on a seat, though, would mean to me something along the lines of: You and your neighbor are inhabiting the same seat. (Grammatically this is not necessarily what it says, but it is what I would understand it as.) Hope this makes sense and maybe helps a little.