If we suppose we and everything in the universe is perfectly flat. That is, we have width and depth, but no height. And we are situated on the surface of a balloon blowing up with air and expanding "outwards". Us flat people can move around on the surface of our balloon but not through it, in much the same way we navigate around the earth, in a plane say. No matter where something is situated on the surface of the balloon, one point is no more central than any other point. In other words, the surface of a balloon (the "universe") has no centre. Is this why we say it makes no sense to say the universe has a centre, or edge for that matter, or is this analogy stupid?
The answer to this question lies in cosmology.
It's because of relativity in positions. Every galaxy appears to be moving away from us and the universe we see is only the observable universe. Further points are more in the distant past, when you observe the deep field you observe the past. This may seem irrelevant to your question, but the idea of a "center" to a universe where any position in the universe, be it from our galaxy or from a galaxy in the Sloan Great Wall is the true center is not consistent with observation. That's because galaxies/clusters of galaxies are moving away from each other, relative to each other.
Every galaxy is slowly being pulled apart from other galaxies, save for those in gravitationally bounded clusters. (The whole architecture of the universe, with massive voids of absolute nothing and bunched up clusters of galaxies reflects this quite a bit) If the universe had a central point, this would be easily observed because there would be variation in the color shift of galaxies that could be analyzed to deduce a central point where the galaxies are moving away from.
That is not what is observed. Instead, pretty much most galaxies (save for exceptions like Andromeda or the Triangulum, who prove the rule since those galaxies are gravitationally connected to our galaxy via the local group) are moving away from us. The Great attractor being not a path to the center of the universe, but the center of the recently discovered Laniakea super cluster. So if there's attraction between galaxies, it's because of gravitational interactions. The universe is also without any real edge unless you count the edge of the observable universe. The observable universe being more a historical record than what the contemporary universe around us looks like and is a byproduct of light speed being so limited. The speed of light in a vacuum is also the speed of information. The whole universe is far more massive by comparison.
So to speak about a center of the universe would require the following:
A region of space that galaxy is moving away from or towards. (Which does not exist, instead relative trajectories and gravitational clustering is all regional at most.)
Edges that are defined enough to pinpoint a center (If such edges exist, they are impossible to observe due to the limited speed information travels in the universe)
Absolute positioning to pinpoint where this center would be. (All positions are relative, the galaxy, the sun and our planet all are moving at extremely fast velocities we don't notice thanks to relativity.)
The centre of something ( anything ) has to be defined in some way by the properties of the something and a coordinate system.
So you can in principle define a centre to the universe, but it only has a meaning in the context of the chosen definition.
You could easily define a different measure that produces a different centre.
And you can define a centre even on the surface of your balloon, simply by arbitrary selection.
An edge to the universe requires a context. How do we define it ? How do we detect it ? The only meaningful measure of this ( in a human sense ) that I know of is "how far can I see ?". That keeps changing.
So at the moment I would say "centre is arbitrary" and "edge is unknown".
Why does it make sense to say the universe has no centre?
We can't be sure it does.
If we suppose we and everything in the universe is perfectly flat. That is, we have width and depth, but no height. And we are situated on the surface of a balloon blowing up with air and expanding "outwards". Us flat people can move around on the surface of our balloon but not through it, in much the same way we navigate around the earth, in a plane say. No matter where something is situated on the surface of the balloon, one point is no more central than any other point. In other words, the surface of a balloon (the "universe") has no centre.
This analogy is often used to justify that the claim that the universe has no centre. But there's no evidence of any higher dimensions. We simply don't know that there's any kind of "asteroids" curvature such that if you keep going one way you come back the other.
Is this why we say it makes no sense to say the universe has a centre, or edge for that matter, or is this analogy stupid?
I wouldn't say it's stupid. But I would say it's presenting hypothesis as some kind of fact. For all we know our universe does have a centre and an edge.
My opinion and 20 Cents:
Humanity did confused many things in history.
Back to the days of the holy bonfires on the squares of Europe, nobody didn't understood that the Earth - is not a center of universe.
We are still believing that speed of light is a maximum speed of information, while there is discovered quantum teleportation with higher speed, infinity speed. Infinity looks much better than 299 792 458 meters per second, is not?
Moreover, we can not describe why does photon getting its super-speed without any boost, immediately, as a part of its existence.
That is why, I'm thinking that we are confused with this great theory about center of universe, which is everywhere, there and here, anywhere...
It is too beauty and too hard to understand, that is why everybody simply believing in it, without tries to imagine anything else.
You could read about Cosmic microwave background. There is a redshift and Milky-Way is moving to the one side of this big-bang sphere.
There is a sphere. There is a redshift. We are inside.
Do you still believe that center of universe is "everywhere"?