Put the mug inside the refrigerator, from a safety point of view, there will be no safety hazards. However, from a practical point of view, there is almost no refrigeration.
The function of the vacuum flask is to keep the temperature of the water in the cup, so it can achieve the effect of heat insulation. If the lid is tightly closed, it will not work. If you just want to cool, you can use a thermos cup to hold the water without covering the lid, but this is unsanitary, so the refrigerated water may have an odor.