> If the conditions at the bottom of two black hole "singularity"s are the same - you would expect parallel, equal universes to be generated each time.

That's a strong no (for any sufficiently complex physical setup with moving parts, falling glasses won't shatter the same way twice).

Two things here are relevant:

1) Lorentz (Butterfly) and Smale (Horseshoe Map) both proved that in some physical systems you can always find initial conditions that are very close (for any arbitary epsilon of "close") that none the less end up far away from each other as time passes.

ie: Unless the initial starting points are absolutely precisely identical without question, then "close enough" isn't good enough to guarentee an identical outcome in the presence of "strange attractors"

2) The Uncertainty Principle tells us that at a fine enough grain (within a certain epsilon) initial conditions are like jelly - you cannot nail them to the wall and declare two systems identical.