Does a water cooler need to be plugged into the wall by itself or can I also plug in my refrigerator into the same outlet

Depending on the voltage of the water cooler you can. But if the voltage or amps of the water cooler is high, it's not a good idea to plug into the same outlet. This can cause overload and/or trip breakers tripping. A fridge draws more amps than you think. And a water cooler can draw a lot of amps as well.

I wouldn't do it.