If matter is constantly being cycled through an ecosystem, does that mean a given ecosystem has a fixed mass?
You can always weigh the biomass of the (global) ecosystem in metric tons, or any other unit, even when this mass would not be constant.

I don't see a reason why the total biospheric mass should be constant (maybe because it reached a saturation limit long ago) on short timescales. On long/geological timescales it is certainly not, as the available carbon, silicate, oxygen etc. for the bottom of the nutritional chains a.k.a. microbes, lichen, fungi... evolves, as the planet itself changes.

That is however not true for the total mass of Earth, which is essentially constant since its time of formation, but the biospheric mass is anyways only a small fraction thereof.