For simplicity, take a cylinder-piston system containing water vapour only.
If you are at saturation and start compressing isothermally, liquid water will condense at the bottom of the cylinder, while the vapour above it will stay exactly in the same state (pressure, temperature and density).
So the cylinder will contain less water vapour in absolute terms, but the amount of vapour of a sample of the gaseous phase (say 1 cm^3) will be exactly the same (= the vapour density is the same).
Repeat the experiment with a mixture of air and vapour and you'll get the same result for the vapour part, but since this time you've got a mass of air as well, which does not condense and is forced into a smaller volume, its density - and therefore the density of the mixture air+vapour - will have to increase, but the density of the water vapour itself will not.
I still have trouble visualizing that two different columns of gas, each exerting the same pressure, will have different densities.
If that can help, the two columns (assuming same volume and temperature as well) will have the
same number of gas molecules, but since the two gases are chemically different, the weight of an individual molecule will be different for the two gases, and so will the total weight - and density - of the column.