The simple answer to your question is that you can’t really create a brush that works like this in blender without creating a custom tool using python. It is possible to create this type of animation in blender, but it is tedious to animate.
You could try creating a grease pencil material that uses a one of those textures as a fill, then draw squares that are 1 blend unit tall and wide. You should then be able to turn on snapping, then duplicate and reposition those squares in the grease pencil object’s edit mode. However, the texture fill settings for grease pencil strokes don’t have an option for disabling interpolation, so they look fuzzy, and it’s difficult to make the scale for the texture match the size of the square.
The second idea I have is just importing each texture as an image plane, then duplicating and positioning those in the viewport using snapping (use alt + D instead of shift + D to save memory since all the planes are the same). This gets around the fuzziness issue because you can turn off texture interpolation when it’s on a plane, but then you will have a ton of planes in your outliner. Animating this would be tedious, but simple enough (auto keying helps). You’d just have to animate the planes in and out of frame (set the curve type to constant to have the block suddenly appear instead of slide into frame).
While experimenting with these ideas, I ended up creating this short animation using the textured planes approach:
Here’s the blendfile I used in case you want to look at it:
example.blend (1.3 MB)