There are a variety of exact black hole solutions to general relativity which describe the spacetime structure within black holes. Because they are exact solutions to incredibly complicated equations, they have been simplified to the extent that they are likely idealizations of ‘real’ black holes.
With that in mind, the highly idealized solution of the simplest black hole, the Schwarzschild black hole, does have 4 distinct spacetime regions. One region is a black hole, another is a white hole, and two of the regions are what we consider ‘normal’ spacetimes. General relativity can’t tell us whether the two ‘normal’ spacetimes are the same universe or two separate universes. In any case, there is no way to traverse between the two spacetimes if they actually exist. And, as @Jeffreyk alludes to, it gets even stranger with rotating and/or charged black holes.
So, while exact solutions of general relativity predict some really weird things, they likely don’t exist in reality