Regarding the 1974 classic How Deep is Whale Shit (link here: How deep is whale shit? - The Straight Dope), I always assumed this saying implied that whale shit sinks to the ocean floor, and thus lies very deep…in the ocean.
Cecil’s answer seems to come from the point of view that the saying asks about the thickness or mass of said feces.
The saying has always made sense to me, even if I misunderstood it.
I seem to recall (but could be wrong) from a whale-watching cruise I took once that whale poo floats - I definitely recall seeing circles in the chop that were significantly flatter than the surrounding waves. It could have been whale pee, though, or something else entirely.