The technical term in rhetoric for "fluff," as you put it, is periphrasis. It's a way of expressing oneself by "beating around the bush" and not "getting to the point."
The examples you cite, however, do not seem to have a superfluity of words that we associate with periphrasis. In
"She has an air about her,"
the word about could be a spatial clue about this "air" she exudes in her demeanor: it is around her, about her, surrounding her.
"There's a sadness to it,"
your use of the word intrinsic comes pretty close, I think, to what the sentence expresses. The word to functions as almost (but not quite) the opposite of about. Instead of the sadness being about "it", it's to it, which suggests to me the "intrinsic nature" of it, as you suggested.
There may not be a rhetorical term for this "phenomenon," but in general terms the "extra words" do serve to give an emphasis or intensification to the "tone," the "air," and the "sadness." The sentences with the "extra" words seem to draw attention to an aspect of the words in a way that the sentences could not without them.
Some other examples of this sort of construction:
His way of dealing with people has a certain edge to it, which I find difficult to tolerate.
She has a mysteriousness about her I find intriguing.
There seems to be a negative tone to the words he uses to describe almost everything other people find quite positive.
The smell of cinnamon has a unique property to it such that the smell transports me magically to my mother's kitchen when I was a kid, with my mom baking her to-die-for cinnamon buns. My mouth begins to water as I imagine myself biting into one of them, and I am transported briefly to Nirvana!
Let's take the "extra words" out of the above sentences to see if their excision makes a difference:
His way of dealing with people has a certain edge I find difficult to tolerate.
She has a mysteriousness I find intriguing.
The words he uses to describe almost everything other people find quite positive seem to have a negative tone.
The smell of cinnamon has a unique property such that the smell transports me magically to my mother's kitchen when I was a kid . . ..
What do you think?