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.

Since I don't know this product, I can only answer with (my) common sense:

Air should be a much better insulator than glass, so even if there's no (good) vacuum, the insulation should work pretty well.

One thing to keep in mind is what is mentioned on the Wikipedia article on vacuum flasks

Heat transfer by thermal radiation may be minimized by silvering flask surfaces

and observing that a normal vacuum bottle isn't translucent. That means that I would expect some loss due to radiation, since the glass isn't covered in any way.

I use a double walled stainless steel mug. It works very well and keeps 500 mls of water hot for more than an hour, and yet still handles like a normal mug. The walls are separated by air, there is no vacuum

I would expect there to be more thermal losses with glass, but I would still imagine it to perform well. I have noticed over recent years the popularity of double walled glass in China. They like to walk around with their tea all day, so it must work well enough

I am actually doing this is school now, the thermal cup keeps it warm, because it is a better insulator meaning it doesn't transfer heat much. So the heat will stay in the cup and not spread out in the air, the air pocket is to slow down the transfer of heat.

I recommend this type of cup, if you are looking for one to take to work, or just keep it warm.

Since there is no thermal break between the inside and outside glass, the heat of the liquid will transfer through the glass as if it was a twice as large cup, so pre heating cup should help.,

Heat is transferred from one object to another by conduction, convection, evaporation, and radiation. #3 is the only one to mention radiation, which is key to the old thermoses I brought to school that have mirrored/silvered glass on the inside. They also shattered if you dropped them, which dramatically reduced their popularity since no one wants to drink crushed glass! The mirrored/silvered surface works by reflecting the radiant heat, which consists of light in the infrared spectrum, back towards the liquid. Think of the night vision goggles used in movies and the military that "see" the infrared heat coming off of your body. The thin, silver, emergency "blankets" sold now work the same way, reflecting your body heat back to you. To keep your coffee warm the longest time, you want to block all 3 pathways. The double wall reduces heat transfer by conduction and convection. A vacuum middle layer would ideally block both these processes much longer, since there are no molecules in a vacuum to transfer the heat. Interesting comments that the double steel wall mugs work better. I think steel is a better conductor of heat than glass or ceramic (which can have tiny "microbubble" air pockets in its makeup). Think of the steel fins in car radiators, air conditioners, as well as laptops and computers, used SPECIFICALLY for their rapid transfer of heat! Lastly, the steam coming off of your fresh mug of hot coffee (that makes it look so good!), is evaporation taking place at the coffee and air interface, as energetic water molecules break loose and launch into the air. Evaporation sucks a whopping 540 cal of heat from your coffee per gram of water evaporated. Much faster than when the coffee looses heat by direct contact with a cooler surface (conduction), which is only 80 cal per gram of water. A simple lid cuts the evaporation way down here. But then it still condenses on the underside of the cooler lid surface, giving its 540 cal to the lid. But, this is still better than losing it all to the air in the room. SO, the best coffee mug would be a double wall of material with the lowest heat transfer rate, WITH a vacuum between layers, AND with a lid. Actually, I failed to mention one other factor. The heat conducted from the coffee, to the bottom of the mug, to the table surface, which is the only place the mug physically contacts another solid material. This is why all coffee cups have a raised lip around the bottom, since this reduces the amount of coffee mug touching the table. To be extreme, you could make your cup with little spiked legs that would reduce the surface contact by 90% or more! To go even further, two inventors shown on the Sharks TV program, designed small cubes of a material that initially absorbs heat from the coffee, and then slowly releases it back to the coffee as the temperature of the coffee goes down. yes, my BS was in Biomedical Engineering. And as a doctor now, heat transfer is key to treating hyperthermia and hypothermia.