It was apparently stamped "USSS" at some point between its date of manufacture and the date of the filming of the video.
At least that one is known.
My recollection is, the Sharps rifles provided for the Sharpshooters were particular, with the double-set triggers and sights to Berdan's standard, etc., so somewhat distinct from the standard M1859 Sharps rifle.
By 1863, as the ranks of the Sharpshooters thinned, the unissued Berdan's sharps and those in store, were issued to other units by the Ordnance Department. This suggests to me there was no military purpose to marking the above rifle with "USSS."