A bit unusual, in that anyone who wants can don a pair. But yet, they still have a “cool” aura.
What’s in it?
I live at high altitude with lots and lots of snow. The sun is very bright. I won’t go anywhere without my sunglasses, and often catch myself wearing them on overcast days or near dusk.
Once visiting relatives in PA, they thought I was trying to be cool by wearing sunglasses. Not at all. It’s just that when you are used to wearing them whenever you are outside it becomes a habit.
Best guess off the top of my head: hiding the eyes makes it easier to seem a bit aloof, which is generally seen as “cool.” (A cool person is largely a bit disaffected by their surroundings.)
I mean, you don’t have eye contact, but without looking like you’re actually avoiding it. You can’t see any nervous twitches. And so much emotion is conveyed around the eyes, for example, it’s how you know if a smile is real or fake.
They also tend to be in cooler shapes to augment this, of course. But I think the base is probably that your eyes are covered, while the rest of the face is still visible.
Concealing eyes = aloof = cool
Also, styles like aviator sunglasses are evocative of cool professions, such as fighter pilot.
Well, not all sunglasses are cool.
They often are useful for hiding/minimizing uncool facial features. There’s a classic photo of convicted killer Carl Coppolino, his lawyer (F. Lee Bailey) and Coppolino’s wife circa 1960, all wearing shades and looking cool (I couldn’t find it, but here’s one featuring Carl and Bailey). Coppolino looked a lot geekier and less reliable without sunglasses.
My mom tells me of being younger and wearing sunglasses and my grandmother would disdainfully say “Who do you think you are? Hollywood?”
The idea of it cracked me up and, decades later, I still break that out from time to time when I see someone wearing sunglasses.
Au contraire mon frère. I love those frames!! And I’m the epitome of cool.
Actually, my Ray Ban Clubmasters are my current favorites.
ETA: to answer the OP, sunglasses just are cool.
The eyes are the window into the soul; sunglasses are pulling the curtains shut. That is just darn cool.
I think we connect and get signals from people’s eyes more than we might think, such that anything that hides the eyes makes someone automatically seem more mysterious, and mystery (especially non-threatening mystery) is “cool”.
How about Venetian blinds?
Yeah ----- I know. But my first though was “cover his head with a Greek Urn”.
Extra points if you know the answer to the follow-up question after that line. 
Another possibility is that part of the reason is that sunglasses are (almost always) intentionally perfectly symmetrical. Symmetry is viewed as a sign of beauty, and this is a great big, symmetrical signal where you normally look when seeing someone.
About 300 euros a week.
Back in the day, my buddy Mark “Whitey” Cooper had a band in Pittsburgh called Whitey and the Blind Venetians.
Those were the days…
Jeez, I don’t know if I’d want to go around looking like J.J. Hunsecker in Sweet Smell of Success.
Some film historian called Burt Lancaster’s specs “the most evil eyeglasses in movie history.”
Hah!! My second favorite are my Wayfarers, but people always mistake me for Nicholson.
Ooh, I want those! I love my Wayfarers, but those cat eyes are awesome.
Almost everyone I know wears sunglasses, so either we’re all cool, or none of us are.
Mirrored sunglasses are the coolest. They don’t merely turn the windows to the soul into black holes, they reflect your own soul back into your head.
The coolest of the cool mirrored sunglasses from Cool Hand Luke.
That’s exactly the style I wear. I love the look. Mine are prescription and I wear them virtually every time I go outside during daylight hours. In the summer I’m sure they get as much or more wear as my indoor glasses.