How does curry shop keep their curry hot without reducing them?

Many curry shops and other varieties of fast service restaurants and take out places will indeed keep a variety of broths, soups, stocks, and sauces hot during service hours.

Other components of the dish are kept cold, and heated when the dish is ordered. Some parts may need to be quickly cooked, then the broth added. A curry shop will do exactly this.

When you order a green curry with chicken, they will take a portion of chicken and some vegetables, sautee them until the vegetables are cooked and the chicken hot, then add the sauce. The sauce might be cold or hot, depending on how busy the place is.

Now, to really get to your question...

If a sauce or soup is kept warm for hours, you are exactly correct, it will reduce and get overly salty. There is a very easy fix!!! Add water !!! In any restaurant I have worked I would add water or stock throughout the service period. I would be tasting as needed and adjusting, but it just that simple. Add water. (or heavy cream for cream soup)

A takeaway lunch place I used to enjoy visiting kept their small range chilled (definitely not room temperature, that wouldn't be safe) and reheated to order. But that was somewhere that was expecting to serve quickly.

More often such dishes are assembled to order from components prepped as much as possible in advance and chilled. If the main ingredients are cooked beforehand, this is quick enough. A glance at the range of options on a typical curry menu would suggest that they'd need many containers of finished curries, but keeping the sauces and the meats separately chilled allows them to be combined into a wide range of options.

Dishes can be kept hot, typically something like 60-80C so not boiling. The low end is limited by food safety (and I'm not up to date on the exact numbers; there's also a maximum hold time) while the upper end is limited more by quality. Sauces can in theory be topped up with water but this isn't normally seen in practice.

I think they use a double boiler type of pan. So with the top pan holding the curry, there is warm/hot water underneath it keeping it hot. If they kept the water anywhere near boiling it would be plenty hot for the curry to stay warm but not boil over.