One of the defining characteristics of both Natasha and Pierre is their impulsive spontaneity, which drives them (on more than one occasion) to make very irrational decisions. While pairing two people with this mindset may seem like a recipe to open up an entirely new book of problems, I choose to hope putting together these two passionate people would satisfy each's desire for love, meaning, and importance.
Another bonding factor these two characters have is the experiences they have gone through. Two strike my mind in particular: Firstly, both Natasha and Pierre have undergone a quick whirlwind of religious transformation (or epiphany, whichever you care to classify it as), and secondly, both Natasha and Pierre are now living with the humiliating gossip and mocking of the many aristocrats that condemn their romantic situations. Pierre's sudden obsession with the Freemasonry and Natasha's recent change into her "better," more religious self show readers that they are both looking for more from life--perhaps they could find it in each other? Additionally, while Pierre has always been the sort of fellow to be mocked by others, it arguably has never been quite so bad as after his unfaithful wife helped to make a fool of him. Natasha too, shares the burden of an embarrassing romance: her near elopement with Anatole and broken engagement with Andrew have undesirably brought the public eye of disapproval upon herself.
Perhaps both characters are too spontaneous for love to ever work out in their favor, but who knows; a girl can dream, can't she?