How dried up detergent water led to precipitation of snowflake like shapes?
There are many water soluble molecules in soap. I believe that as the water evaporated, the concentration of these molecules became great enough for them to start crystallizing. This process is used in chemistry as a means of purification and is known as recrystallization.

Are you considering the X type shape of the crystals to be snowflake like? I am not quite sure why they all seem to have similar geometries, I want to guess that they are crystals of citric acid.