As others have pointed, mass is invariant and easy to measure in any planet, with a balance scale and a set of weights or even with a kitchen scale set to be used in a given planet gravity.
However, measuring mass underwater (or under other dense fluids) is difficult because objects experiment buoyancy, and buoyancy is the product of density of water and volume of object.
For objects of known density, that problem can easily be overcame with a little maths. Let $measured\ weight$ be the weight measured by a scale underwater, given in units of mass ($scale\ result$), calibrated to local gravity (in Earth a waterproof kitchen scale would be fine):
$$scale\ result=g·measured\ weight=weight-buoyancy=$$
That is, to determine mass underwater you just to multiply for a factor depending of density of object and water. For example for steel (density=7,85 g/cm3) under fresh water (density=1 g/cm3), we only need to multiply what the scale says by 1.146.
Mass of objects with unknown density are very difficult to determine underwater. For heavy materials a quite rough approximation of density could be enough but for materials of density close to that of water (like wood or kelp) an small uncertainty in density would cause a larger error in mass. Therefore, a lot of commodities that in our land dwelling world are measured by mass are likely to be measured by volume or by units for underwater trade.