I checked the American Association of State Highway and Transportation Officials (AASHTO) records and found that both bridges are indeed designated as U-S 23 Spur (since 1986), dead-ending at U.S. 52 on the north end of the bridge.
The link is: http://www.us-highways.com/spurus.htm
But, the U.S. 23-S designation is not used and may be truncated permanently, because of, as you speculated, its confusion with U.S. 23 South. Since there are no signs for U.S. 23-S in Ashland (I didn't spot any when I took the pictures), there isn't any confusion.
But AASHTO indicates, neither bridge carries U.S. 60.
The National Bridge Inventory reports that this bridge carries "US 23S" and classifies the route as a "U.S. numbered highway spur". Presumably the "S" stands for "Spur" and not "South". Bid documents by the Kentucky Transportation Cabinet also refer to the bridge as carrying US 23S. For example:
Moreover, the route number of "23S" appears in the bridge's inventory number: "010 023S B00040".
Great website, very useful information. I have a correction for you, however. This bridge is NOT on US-23, but is actually on US-60. Specifically on 13th Street. You can check a map closely, and there is a photo at http://www.kentuckyroads.com/images/ohio_river/.