If hes using water magic... wouldnt the refrigerant used be water? Which seems kinda useless given the temperature ranges water is gas at, not even sure with some vacuum it can be gas at useful temperatures.Refrigerant is not the issue there are many common refrigerants, like ammonia, propane.
The roadblock to refrigeration is the ability to produce, compress and store gasses. The first of what we would call a real air compressor was in 1829, people had been experimenting knowing that the evaporation of liquids could drop temperatures such as evaporating solvents under vacuum since 1804 and some experiments earlier. It wasn't until the invention of the air compressor that the first working compressor based closed loop refrigeration cycle was made in 1834
In addition, idk if the translation was wrong, or the author was wrong, but refrigerant does not become fully liquid during cooling or compression in most cases and he just needs an expansion valve (small hole) to turn the liquid into really cold gas
Seems his refrigerator works with a gas and a liquid phase, starting on liquid otherwise his water magic would be making gas. Though at that point you might be able to skimp and just vaporize the liquid and then send it out to the atmosphere and make more liquid and avoid compressing it to reuse it.
If water magic can make other liquids, isn't that already wayyyyyyy more busted than some janky refrigerator?
Also... "Wind magic makes the released heat work it way inside the refrigerator" Is literally the opposite of what you want.
There might be some issues with translation or the guy who is adapting it to manga... probably got glossed over pretty hard in the original novel