Sorry, I should have provided references but, as I said, I wasn’t sure I was even on track with your OP.
For Kierkegaard, I had in mind the
Concluding Unscientific Postscript, Part Two, Chapters I, II, and III. I won’t even pretend to give a summary! But his stance, briefly, is that Hegel’s philosophy (which was very much in vogue at the time, and which SK calls “the System”) can give an account of absolutely everything except SK himself. He is left understanding all that can be known (he’s being a little sarcastic here) but the one thing he wants to understand best, his own being, is left out of “the System.” And this must necessarily be so, since there is no room for anyone’s subjectivity in an objective account of what is. The “uniqueness” angle might be that “the System” treat only things which are
not unique, because they must be shareable and rationalized. Whereas any given self is, arguably, a one-off, a
hapax – so how can it be made an object of knowledge?
The Nagel reference is more specific. In
The View from Nowhere, p. 54, Nagel writes of what he calls the “centerless” objective world:
Yet each of us, reflecting on this centerless world, must admit that one very large fact seems to have been omitted from its description: the fact that a particular person in it is himself. What kind of fact is that? What kind of fact is it – if it is a fact – that I am Thomas Nagel? — Nagel
So again, subjectivity is posed as a problem for the objective viewpoint. (The puzzle is only a puzzle when expressed in the 1st person.) I think the “uniqueness” question here is more or less the same as in Kierkegaard – Nagel is uniquely Thomas Nagel, and this is of extraordinary importance to him, not in some egotistical way, but simply as a plain matter of getting around in his life. How do we fit such a vital fact into our “System” of non-unique things?
Hope this helps.