That "border" is merely anti-aliasing.
Draw a vector shape.. export or save it to a raster format, the anti-aliasing will be introduced.
You could also use the Shape Tools to draw a raster/pixel shape and the anti-aliasing will be introduced the moment you let go of the mouse.
I am not sure you are planning your print the right way.
As I understand your project will be printing black and green on the cardboard:
So you should separate your files, a plate for the black and a plate for the green.
But if you want white over green, you need a white plate that will look simply the same as the black plate, using white opaque ink. There ...
First, to select without white line, the best way here is to select with a rectangle with precision by hand (zoom in). Once a plank is selected, right clic on it with a select tool active and Layer via cut.
Do it for each plank (you can move your selection instead of doing a new one since you seems to have multiple same size here.
Once you have each plank on ...
Hmmm... It certainly appears as though overprint is set for the blue type. But your layers aren't showing that appearance meatball. Groups won't show appearance attributes in the Layer panel unless appearance is applied to the group itself.
One thing is certain, the issue must be the type layer...
Check inside those three groups for the type in your Layer ...
I too ran into this bug in version 2.10.22. Per the bug tracker it should be fixed in version 2.10.24, which is not yet stable at the time of writing this. Here is the link to the bug tracker item.