Sort of a complicated question, I'll try to ask. How was I born me and not another?

[FatAndFaithless]

Again, if you're just talking about the potential combinations of your parents' genes, that's something we can work with, because that's a quantifiable number.  

But saying "You could have just as easily been the egg" doesn't make any sense still.  You're not now nor were you ever an egg or a sperm.

I think the question still remains though. Because there's so many possibilities of other people being born from the same mother, it seems improbable that you specifically were born from that mother, or anyone for that matter. That's not even to take into account the billions of years of evolution leading up to you being born. I just can't fathom how improbable my birth actually is.

I'll ask again.  Is your question simply about the possible genetic combinations of your parents (and further ancestors)?  Because that could be conceivably quantified (if not practically, since we don't have access to the genetics of more than a generation or two back).

Yes, the specific combination of genes that is now "you" is one out of a gargantuan amount of other possibilities.  The odds that your parents gave birth to your particular genome is very tiny.  Not sure what the use of knowing the specific odds would be though.