That photo of the Hubble Deep Field really messes with me... in fact any photo like that. You have to realise: somewhere out there, in fact, most likely, many, many places out there, are planets where everything we have and will experience has already happened. Whatever end-point human evolution is driving towards (massive environmental degradation, simulacrum-building, rise of a superhuman elite, space-travel, true self-knowledge, total mastery over the physical world, whatever) - its already happened. Over and over again. By people/things we will never meet and never know about.
If it's uniqueness you're concerned about, there are a lot of different parameters that describe habitable worlds and even more to describe the history of a civilization. Odds are, what most civilizations experience is unique to them; but I too believe there are likely to be many similarities, too.
Our inability to see and interact with other civilizations - even within the relatively small neighborhood of our own galaxy - is frustrating, and somewhat of a logical conundrum as described by the Fermi Paradox. Whatever the factors preventing us and others from making contact are, and I think it's likely a combination of several instead of one big filter, I hope we can leave them behind at some point and join up with the other people out there.
When I look at bigger parts of the universe, I feel inspired. The universe is not small and limited, it's vast and full of possibilities. Odds are we're not singular but in good company, and we have a lot still to discover and explore. What we do and know is unique and meaningful, yet at the same time it's not all there is.
I have completely different feelings about this image, but I think you are correct in attaching a lot of meaning to it. We're very lucky to have been born at a time were we get to see it.
You're experiencing the effects of Douglas Adams' Total Perspective Vortex (from Hitchhiker's Guide to the Galaxy).
It was basically a torture chamber which inflicted the most horrific sort of torture imaginable upon its victims. How?
"For when you are put in the Vortex, you are given just one, momentary glimpse of the size of the entire unimaginable infinity of creation along with a tiny little marker saying, 'You are here'."
When the victim sees his complete and utter insignificance, his psyche is utterly destroyed beyond recovery.
Feels very disheartening.