MgO as food supplement - solubility in water
MgO will react with stomach acid, not dissolve in it. I suppose that's what the manufacturers go for.

This is not factually right at all. Stating the metal content as an oxide is an century old tradition of quoting the metal content as oxides. Magnesium percentage quoted as percent MgO does not mean that the tablet actually contains magnesium oxide. The links OP provided in the comments shows that the tablet actually contains magnesium stearate. This is done to avoid upsetting the stomach because a very water soluble magnesium salt can induce diarrhea in a few hours.
Magnesium steareate is a salt of a fatty acid, which is rather water insoluble as expected but it has some fat solubility. So when you take it with food which has some fats along with this fatty magnesium salt, there must a be slow absorption of the fatty salt by the body hence the tummy is happy.
Basically, you are eating a magnesium soap!