I recently read that math/physics for a holographic black hole could be applied to the entire universe,
I'm afraid the holographic principle is a mathematical conjecture that's more than twenty years old, and for which there is no evidence whatsoever. In fact, since all the evidence and experience says we live in a 3+1 dimensional universe, I'd go so far as to say the holographic principle is a "lost in maths" conjecture.
So if a black hole is a 2-D structure
Who says it is? We're confident that massive stars collapse to form stellar black holes which have a spherical symmetry. They aren't shaped like a circular piece of paper of zero thickness. They aren't two dimensional. Yes, one can think of a black hole as per Oppenheimer's original frozen star and liken it to a gravastar which features "a void in the fabric of space and time". But it's only a spacetime "void" in that light can't move, because the coordinate speed of light is zero. It isn't really a hole in space with a surface, it's more like a hulking great lump of matter-energy in a form that we don't usually see. It's no more two-dimensional than an iceberg is two-dimensional, regardless of what some fish might tell you.
Does anything interesting arise out of that?
The universe as a whole can be likened to a black hole in some respects. That's interesting. But neither are two-dimensional.
What does it "look like" when a flat surface 'collapses' out of 3-D space (a black hole) onto the 2-D cosmic horizon? Or does the 2-D surface of the black hole somehow remain in 3-D space?
You could argue that it looks like a hole in space. Think of a balloon, like the sort of balloon you have a at a childrens' party. Imagine it's bluish grey, and speckly. Now look at this picture.
CCASA image by Alain Riazuelo see Wikipedia
It looks like a balloon with a hole in it. It looks like a hole in space. But it isn't really a hole in space. It's more like a big cold lump of solid energy, a bit like the iceberg. It's massive. Or even supermassive.
Trying to boil this question down, is there anything interesting about a nested holograph? Does anything interesting arise A 2-D holographic black hole inside a 2-D holographic universe? Or is it just a 'huh'?
No and no. It's just a "huh". And if anybody begs to differ, you will be able to read why in their answer, whereupon I will humbly apologise for my ignorance. But don't hold your breath.