"Floor climbing" activity always counts 0 floors climbed

Are you going up physical stairs, or just using a star stepper?  The watches uses an altimeter to measure stairs, I don't think it would count a stair stepper.  That said, it's strange to me a built in stair stepper app wouldn't count steps as stairs!  Maybe worth submitting a bug report to Garmin?

Physical stairs

The stair stepping profile is intended to be used with a stair stepper machine. If you are climbing physical stairs you should use the Floor Climb profile.

I'm sorry! I have the Italian version and I didn't know how the name of the app was called in English; I did a litteraly translation from italian but I was wrong.

I used the app called "Salita scale" with a man climbing a three step stair as icon. On the watch display time and floors climbed are shown while Connect reports the total ascent and total descent.

Therefore, I believe I have used the correct app - "Floor climbing" (so not "Stair stepping"!) - for climbing stairs in my building but floors are not counted despite a total gain elevation of about 25-30 meters.

Ok. Then it is correct app.

Do you get any ascent and descent in the activity?

No; the few times I have used this app, I have used it continuously climbing up the floors of the building where I live. At the end of each floor there is an horizontal section before starting the next flight of stairs.
In total 6-7 floors, depending on the times, but the total marked on the clock has always been 0.

The first thing you need to make sure is that you get ascent measured. Floors is just 10 feet of ascent.

Does the watch count floors during the day?

Add the Altimeter widget to the widget loop and check that you get good altitude measurements.

Auto should work. The absolute value might be a bit off but the difference should be correct.