The bug
The banner marker icon is moved by one pixel relative to its correct position when reloading the world after adding said banner to the map. This can only be manually fixed by removing and re-adding the banner to the map. Using icon textures with the icon perfectly centered on the spot the banner is located makes this issue pretty obvious.
Note
The vanilla icon texture behaves the exact other way round: It is misplaced by one pixel when adding it to the map but correctly centered after a world-reload. So it might be possible that maps actually show the incorrect banner position when first adding them, but updating to the correct one after a world-reload.
If this isn't being fixed for 1.13.1, it would at least be good to know in which case the correct marker position is shown, so I can adjust it in the texture file if necessary.