BUT, have mercy - what a heck do you expect them to look like? It they really were randomly (not pseudo-randomly) selected - they would look exactly the same. A square here, straight line there. It's just the consequence of the fact that you use only two symbols. Use six, then we'll see what is the randomness.
Mmmm...No, the main point of my little puzzle is about Pareidolia, the human tendency of seeing "faces in the clouds".
Most thinkers find purpose on nature, see the patterns, the mechanism, and deduce that a creator have designed it, or that a deep mistery about nature exist.
However, as humans we are condemned to find patterns anywhere, to match every information with our existing prejudices. I think that small labyrinth, which in reality amounts to a bag of stochastically (not random) distributed bars, show that lesson quite nicely.
The question about randomness is a secondary point. Most idealists try to scape the consequences of determinism and modern neuroscience through quantum mechanics. Most physics thinks that indeterminism exist at microscopic level, and others think indeterminism is NEEDED for QM and General Relativity to be correct.
However, as you pointed out: How can one find a diference between true randomness and stochastically generated noise? And, even if you can, does it matter? Would our universe, and our own nature as humans, be different if indeterminacy exist at any level? I don't think so.