If my father and mother had not had together....?

If my father's sperm had fertilized a different woman's egg instead of my mother's would I still have been born, but looking different due to having the genetics of a different mother, and if a different man had fertilized my mother's egg instead of my current father, would I still be "me" but just looking different because I would have the genes of a different father? So then, if my parents had not had together, but instead they had paired off with 2 entirely different people to creat children with, would I then actually be 2 different people instead of the one that I am??