That is my first reaction as well.
I assume that's not what the OP meant though. From his post, I assume he just means there's a left-right balance issue.
You can kind of see the issue in the photo. He has a heavily dropped right shoulder, but the hem (as is) is aligned. There's also some slight pulling just under his right lapel. My guess is that -- if he were to unbutton his coat -- the right side would fall. What he mistakes as a misplaced button is probably just a non-corrected right drop.
This happens because tailors cut the left and right panels out together, with the fabric folded over onto itself. If a client has a heavily dropped right shoulder, they need to adjust for that, but not all do. So when you lay the coat flat, it looks fine. When you wear it, however, the button and buttonhole might not align. That's an easier thing to spot when you're just throwing the jacket on than scrutinizing to coat to see whether the hem is balanced.
If that's the issue, it's pretty easy to correct at the shoulder seam.
He could also move the button -- but that only fixes the button alignment and not the fundamental issue of a bad left-right balance.
First of all, thanks for this very detailed answer.
You are right in that the buttoning of the jacket has caused all sorts of distortions, and I have included above a picture of it unbuttoned. I think the jacket fits much better, although the left-right balance issue still persists.
I will work with the tailor, and will be reporting back here for all's benefit.