How long have the club soda and the comparable water been sitting out while exposed to the open air?
Lets look at an open container of club soda vs. an open container of pure water. I will assume this club soda doesn't contain sodium bicarbonate, and all it contains is water and carbonation.
The CO2 when in contact with water forms carbonic acid. The colder the water and the higher the pressure, the more CO2 you get into the water and the more carbonic acid forms. Open the container, and the pressure drops, some of the carbonic acid goes back to CO2, and CO2 flies out of your club soda container. But, not all of it comes out, and it doesn't all come out instantly. There is a water / CO2 / carbonic acid equilibrium that is eventually reached.
The club soda will taste flat far BEFORE all of the CO2 comes out. So, even at that point, it'll be water + CO2 + carbonic acid in one container vs. water in the other container. So, no, initially (flat club soda) != (water). That symbol means "does not equal".
However, if you give it enough time, the club soda will reach an equilibrium with just small amount of carbonic acid. Also, with enough time CO2 will ENTER the pure water container and also it will reach an equilbrium with small amounts of carbonic acid. Thus eventually, (really flat club soda) = (really flat water). However, really flat club soda will never be pure water.