Does a double walled glass mug really keep the liquid warm?
Yes, they work.

The reason they work for keeping liquid warm is because the air pocket slows down the transfer of heat from the liquid to the glass to your hand. Air has a lower thermal conductivity than glass does, which means that it slows down the loss of heat from your drink. (The thermal conductivity of air is 0.024 W/m/°C, while the thermal conductivity of glass is anywhere from 0.96 - 1.3 W/m/°C, depending on the type of glass.) It's the same reason that windows in modern homes are usually double-paned.

The space between the layers of glass are not a vacuum.

That being said, glass double-walled thermoses are popular mostly because they look cool (they give an optical illusion of the liquid being suspended in air). If your primary concern is keeping your beverage warm for as long as possible, it's better to go with a double-walled stainless steel thermos, which will retain heat much longer than a glass one. MargeGunderson is right, too - preheating the thermos, regardless of material, with hot/boiling water for a few minutes before putting your drink in it will be even better than pouring your tea into a room temperature thermos.