I think it was used in the sense of "dead as ..." or "meaningless as..." or "useless as..." But in this instance (story below), it seems to be close to "dead as..." but not quite. I wouldn't think "stiff upper lip" is exactly the same thing—that's more like stoicism in adversity. In this instance, Grant was receiving a note that Lee wanted to surrender, which strikes me as a bit different mood—one you'd expect to be exultant, or filled with testosteronish backslapping.
I remember first hearing this about General Ulysses S. Grant, in a quote in Ken Burns's The Civil War, but it was in much more common use in the nineteenth century generally (not just the Civil War period) with a variety of meanings.
Here's what Google Books has to say: The gentlemen have just received a note from Confederate general Robert E. Lee; it is April 9, 1865.
"There was no exultation manifested—no sign of joy—and instead of flushing from excitement, he clinched his teeth, compressed his lips, and became very pale. Grant read it through mechanically, and handed it back to Rawlins, saying in a common tone of voice, 'You had better read it aloud General.' The immovable expression of countenance in these two prominent actors in the great drama drawing to a close, was rather discouraging to the onlookers. Rawlins showed nothing but extra paleness. There was no more expression in Grant's countenance than in last year's bird nest. It was that of a Sphinx."
—Sylvanus Cadwallader, Benjamin P. Thomas, Three Years with Grant (University of Nebraska Press, 1996), 322