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.