Your question is impossible (what now? Same amount of air plus 1 kg water vapor in the same volume and pressure as with 1 kg liquid water?). The question also appears to contain its own answer: "I acquired two boxes of the same dimensions and same weight. ... measure their respective weights ... what about the box containing steam?" Yes, what about it? It has the same weight, post ends here.

Ok, let's assume you meant something else — perhaps mass, not "weight"?

But whatever your setup, the answer is always very simple: The scale will measure the gravitational force the earth exerts on the gross mass of the box¹ (i.e. packaging plus content, including all gas), minus the buoyant force of the box in the atmosphere. Packaging and buoyancy are identical for identical boxes, so any difference in "weight" (= force on the scale) must be from differences in the mass the boxes contain.

Now make up your mind: What's in the boxes (gases, fluids, solids)? Then you have the answer.

_{¹ Or, if you want, the gravitational force that the gross mass of the box exerts on the earth. The attraction is mutual; it's just that the earth cares less.}

Your question is impossible (what now? Same amount of air plus 1 kg water vapor in the same volume and pressure as with 1 kg liquid water?). The question also appears to contain its own answer: "I acquired two boxes of the same dimensions and same weight. ... measure their respective weights ... what about the box containing steam?" Yes, what about it? It has the same weight, post ends here.

Ok, let's assume you meant something else — perhaps mass, not "weight"?

But whatever your setup, the answer is always very simple: The scale will measure the gravitational force the earth exerts on the gross mass of the box¹ (i.e. packaging plus content, including all gas), minus the buoyant force of the box in the atmosphere. Packaging and buoyancy are identical² for identical boxes, so any difference in "weight" (= force on the scale) must be from differences in the mass the boxes contain.

Now make up your mind: What's in the boxes (gases, fluids, solids)? Then you have the answer.

_{¹ Or, if you want, the gravitational force that the gross mass of the box exerts on the earth. The attraction is mutual; it's just that the earth cares less.}

_{² Perhaps it's the buoyancy that confuses you. After all we seem to know that a volume containing as little as possible (hot air, helium, a vacuum) experiences the uplift we call buoyancy, but heavy objects like rocks or tanks do not; and the break-even point appears to be when the density of the contents is the density of air. But that's just our everyday approximation: In reality, all objects immersed in gas or fluid experience buoyancy; it's just that it is irrelevant to our everyday handling of heavy things. The amount of buoyancy exclusively depends on a body's volume (its displacement). It is the effect of pressure differentials between the underside and the upper side on the outside of the body; what's inside is irrelevant (and in the thought experiment of a perfect "black box" unknowable). If your boxes' contents was unknown, and they had the same mass (and mass distribution), they would behave exactly identically in all aspects. Whether the contents is a fluid or vapor or neutronium is unknowable and irrelevant.}

Your question is impossible (what now? Same amount of air plus 1 kg water vapor in the same volume and pressure as with 1 kg liquid water?).

But the answer is actually very simple: The scale will measure the gravitational force on the gross mass of the box (i.e. packaging plus content, including all gas), minus the buoyant force of the box in the atmosphere. Packaging and buoyancy are identical for identical boxes, so any difference in "weight" (= force on the scale) must be from differences in the mass the boxes contain.

Now make up your mind: What's in the boxes (gases, fluids, solids)? Then you have the answer.

Your question is impossible (what now? Same amount of air plus 1 kg water vapor in the same volume and pressure as with 1 kg liquid water?). The question also appears to contain its own answer: "I acquired two boxes of the same dimensions and same weight. ... measure their respective weights ... what about the box containing steam?" Yes, what about it? It has the same weight, post ends here.

Ok, let's assume you meant something else — perhaps mass, not "weight"?

But whatever your setup, the answer is always very simple: The scale will measure the gravitational force the earth exerts on the gross mass of the box¹ (i.e. packaging plus content, including all gas), minus the buoyant force of the box in the atmosphere. Packaging and buoyancy are identical for identical boxes, so any difference in "weight" (= force on the scale) must be from differences in the mass the boxes contain.

Now make up your mind: What's in the boxes (gases, fluids, solids)? Then you have the answer.

_{¹ Or, if you want, the gravitational force that the gross mass of the box exerts on the earth. The attraction is mutual; it's just that the earth cares less.}