Could a gaseous moon exist in the same way as a giant gas planet? All the moons in the solar system are rocky, or icy. Why shouldn't gas planets have gas moons?
LocalFluff's comment is spot-on. You need mass to have gas, and moons just don't have enough.
It is thought that gas giants gathered the gas they have today by accreting large amounts of it from the protoplanetary disk in the early days of the Solar System. At first, they were only gas-less cores (not rocky, exactly, but not gaseous), but they quickly became the most massive objects in their immediate vicinities, and thus gobbled up all the gas nearby them.
Now, less massive bodies, like Earth, can still accrete gas. However, they have difficulty retaining it. Lammer et al. (2014) calculated that under conditions like those experienced in the young Solar System, planets of one Earth mass or less could retain captured hydrogen envelopes for no more than ~100 million years - a long time for us humans, but shorter for astronomical objects.
The hydrogen escapes via atmospheric escape, which happens when particles in the atmosphere have velocities higher than escape velocity. This phenomenon is known as Jeans escape. Atmospheric escape happens more often with lighter gases, such as hydrogen. It is also influenced by stellar winds, which can cause even gas giants (generally close to the star, such as hot Jupiters) to lose some or all of their atmospheres.
I wouldn't call gas moons impossible, but certainly very unlikely.