I think this is a problem with the fitness app, maybe combined with a "feature" of the health app. Just to test, I made note of the number of flights I had climbed as counted by the heath app, then climbed three flights and verified the health app properly added three flights. Check. Then I started the Stairs workout on the fitness app and repeated. It tracked time and calories, but nowhere in the app does it show flights climbed. I stopped the Stairs workout and verified the health app flights counter hadn't changed.
I will find a friend with an Apple watch and see if they can repeat to make sure this isn't a problem solely with my app (which presumably anyone could try to replicate).
Another quirk... I noticed the Stairs workout also doesn't seem to chart my heart rate. I usually don't pay much attention, but the other workouts I often use (Elliptical and Outdoor Cycle) do chart heart rate. All three give a numerical Avg. Heart Rate, so even the Stairs workout is apparently tracking properly, it just doesn't show the graph. Very odd.