I just tested this on my Ultra 2, climbed 2 sets of stairs, watch and iPhone with me, both were recorded in the Health App/Activity/Flights Climbed, and both show up as a data source.
Then I took off the watch and took only the iPhone with me, climbing the stairs.
The Health App did record the iPhone as a source and added the climbed stairs, but only to the iPhone statistics.
And in the Fitness App I can only see the 2 sets of stairs, the Apple Watch recorded. At least at the moment.
Did you already unpair the watch form the phone and connect it again after that?
Unpair and erase your Apple Watch - Apple Support