Shouldn't that read "for a given output power" ?
No - it's the mass flow which is constant for a given power output.
As you climb the mass flow further reduces because the intake air pressure reduces. Openning the throttle offsets this pressure reduction by causing less pressure reduction across the throttle.
I'm sure that this is true but this suggests that you're
maintaining manifold pressure as you climb by opening the throttle, as is the procedure when you have a constant-speed unit at your disposal.
In the fixed-pitch case, 2300rpm produces less power at a higher density altitude than it does at a lower density altitude and therefore the manifold pressure must be less (even though we don't have a MAP guage to tell us this). The MAP is less but as has been discussed the throttle is further advanced for 2300rpm at the higher density altitude.
So in the CS case you advance the throttle to maintain the MAP as you climb but with a fixed-pitch prop you advance the throttle to maintain the RPM but the MAP reduces (less power at the same RPM).