Now, our old buddy Buddha would say that, because there is no specific attribute of yourself that is "you", that therefore there is no "you". Personal-ness is an illusion (I've noticed that a lot of stuff in Buddhism is an illusion. Makes you wonder about Buddha himself.)

The Enlightened One goes on to say that, since there is no you (simply attributes of yourself that can be mixed & matched), you will therefore not be the same person after going through a reincarnation. After all, there is no "same person" for you to be. Whether or not "soul" as such exists is immaterial. Even your soul is not you.