the stain is yellow, suggesting a leak, and an attempt has been made to hide it by replastering and roughly painting. It is not a condensation patch.
If the damp patch is semi-circular (which it is, roughly) then the source of water is probably about the centre of the circle. As the washing machine is on the other side, it might be a pipe in the floor or wall. Is it a concrete floor?
Take up any floor covering in both rooms and see if the floor is wet. Puzzle out the routes of the various pipes. Please post more photos on both sides of the wall, including the pipes and any nearby drains
I think the wooden floor in your pic is already damaged. Is it laminate or old parquet? You can lift parquet, one block at a time. It might be stuck down with bitumen.
pencil a straight level line 100mm above the floor. You can chip off all the plaster below this line, because when you fit new skirting, it will be hidden. This will expose the brickwork, and you may see more clearly what is happening. Exposing the brick will also allow water to evaporate from the surface.
It is not "rising damp" if such a thing exists because it is not all along the wall, but it is a damp spot with a centre.
I think it's probably a pipe leak.