So this is probably not the perfect solution, and something I tried avoid doing, but anyway:
When I run the game, I noticed that, unlike in the 3D view of the editor, the brick pattern is going down (wrong orientation) on each side of the cube.
I actually had the same thing in Blender, but to fix that, there is an option under the texture properties, in the "Image Sampling" section, called "Flip X/Y axis", which did the trick.
I did not see anything similar in Godot under the material settings, so I resorted to the only other thing I can think of doing, which is to rotate the texture by 90 degrees.
Now, when the game is running, at least the texture is aligned the right way on all sides.
I am not sure this is the best solution, so if you have any better ideas, please add them.