Neji and Hinata are cousins not real brother and sister
Neji and Hinata could be seen as brother an sister, because of their respective fathers. Hiashi and Hizashi were twins. So genetically speaking, they have the exact same DNA. What this means for Neji and Hinata is, the only genetic difference between the two is that they have different mothers. It would be like a man have children by two different women, but the children would be siblings.
Plus Hinata and Hanabi affectionately called Neji, Nii-San. So they were close regardless.