That is a periodic tiling (just think about what happens if you follow the tiling along the rows instead of along the columns, you'll periodically see the same pattern).
It's listed as a periodic tiling. If you move the whole plane one square to the left or right you'll find that everything fits into place nicely, making this a periodic tiling.
Hm, I'm not knowledgeable about this subject but this seems like the offset on each row is NOT random, but it's instead repeating. Imagine instead that each time I added a new row, I chose the offset to be, say, a completely new value.
If you allow reflections of the single tile, an easy counterexample is the pinwheel tiling[1], which is a non-periodic tiling composed entirely of isometric triangles.
An (imo less satisfying) example which does not require reflection of the tile would be as follows:
Take a rectangle with width equal to twice the height. You can use this tile to create squares which are either split vertically or horizontally. Put a single horizontally split square at the origin, then tile the remainder of the plane with vertically split squares: this tiling is (rather trivially) not periodic.