Not a physicist but that's how I understand it, anywhere you go in the visible universe should have a perspective that looks more or less just like ours, as if they were at the center. We can only see so far though, so someone a billion light years from here would have a visible universe that overlaps with ours like a Venn diagram, but would be seeing things we can't. Who knows, maybe there's something really interesting they could see that we cannot, like an edge of the actual universe with nothing beyond?

I think it said right on the map that it's not only essentially the same in all directions, it's pretty much the same from any vantage point, since the distance makes most of the difference in what would be seen.

We are the center of the Universe. Of our observable universe, at least, by definition. And the total Universe might be much bigger, potentially infinite [or even smaller than the visible universe!]. Even if finite, the Universe would have no center. It's different from a finite amount of matter forming a sphere within an infinite grid, but much like a spherical surface that has no center (the sphere does, the surface does not).

> Would this graph essentially look the same if the observations were made elsewhere

Yes, an alien in a very far away galaxy would see a similar picture. If they are within a few billion light-years, they would see a younger red-shifted Milky Way.