How do they refill their soy sauce or chill sauce in Asian restaurants?

You guess it right, they keep topping it up. The same way those fast food restaurants and all other restaurants, they keep topping up their condiments. This may shock you, they top up their soda fountain too and those soda fountain tanks are never washed. It's impossible to wash anyway. Because the tanks are conjoined. The flavored drinks comes in powdered form never premixed and mixed with water in the kitchen and pour into large plastic bottles and top up as needed and never in plastic bags, too difficult to pour or storage. The carbonation cartridges are recycled. Ice cubes and plain water are same. That's why the soda from the restaurants soda fountains taste better than your can or bottle soda. '

Yes they keep adding to the stuff left in there. Soda are in plastic bags and they are used until empty and then replaced. At least in the US it is that way. They have not used syrup tanks for 25 years in most areas.

Cecesan, hate to tell you but the soda fountain syrup comes in plastic bags where I am from and those are tossed when empty. The CO2 cannisters are returned to the company for refills when empty. None of that is topped off.
In a restaurant, it is true, the ketchup and mustard is topped off but when they are totally empty is when they are taken to be washed or washed at a certain schedule.
In places that use bottles of condiments the half ones are poured together or if there isn't much left, tossed.
Most places go through that stuff so fast it has no time to spoil.