It may not always be so, we do well to remember that all the answers we have are provisional, they stand until new or further evidence comes in, until we gain the ability to understand better than we currently do.
But, okay, to actually answer this question: it seems we have SRGAP2c to thank:
"Cortical development gene Slit-Robo Rho GTPase-activating protein 2 (SRGAP2) has been highly conserved over mammalian evolution, and human is the only lineage wherein gene duplications have occurred (three times)."
So, basically, as I understand it, it fosters this neural hyper-connectivity which is lacking in other species despite the fact that we share something like 99% of our DNA.
Yes, I believe the human mind is unique. The human mind has millions and millions of possibilities. It can create new ideas, carry out many things and dream up dreams we never thought were possible. The human mind is an amazing thing and very different from animal minds or other things. We have cognitive abilities and that's pretty amazing.
While there are a wide variety of intelligent species on earth, the advancements humans have made for their society that no other species has aren't all mere coincidence. The human mind has shown an ability to evolve, constantly ask "why" and go about determining the answers. We are flawed for sure, but also very unique.
The human mind is very unique, I think science and our creations have proved that. We have managed to do some really remarkable things in life that could not be described as anything, but unique. Each individual is unique as well, we all have our own thoughts and ways of getting things done.
The human mind is a possible derivation of what is required for our type of intelligence (dna stored information).
It may be unique to humans on Earth, but not to the Universe. Seeing as it is possible to store information in other types of DNA storage, is safe to assume that the brain has many variations, ours being one.
Although seemingly unique, it seems to be because it can learn at a faster pace than all other intelligent life form on Earth, this does not mean it is unique, just more advanced than what we are limited to knowing in comparison.
The brain is an organ. Like any organ it can be created by Evolution, thus not unique, just rare.