Yes, travel mugs keep drinks cold in the same way they keep them hot.
How they do that differs depending on the construction, but the basic principle is always the same.
Heat or cold is transferred through the air, so all cups use some kind of material between their contents and the air to keep the temperature as steady as possible.
The simplest cups use only one material. They have a plastic, glass, ceramic or metal wall and nothing more. This does insulate your drink, but not well. Hot drinks cool fairly quickly and cold drinks heat up quickly.
Better cups use a second layer. The may have two ceramic walls with space in between, or a double layer of metal or plastic. This does a much better job of insulating, but if the space between the layers is filled with air, heat or cold is still transferred, just at a slower rate.
Filling the space between the layers with a more insulating material than air improves things considerably. Styrofoam works well, but the best material to use is actually nothing. Literally nothing.
A vacuum.
Since air transfers heat or cold, the complete lack of air means that cold and heat can not be transferred.
As a result, hot liquids stay hot much longer and cold liquids stay cold. Vacuum tumblers cost more, but they do a much better job of insulating and anything inside will retain most of its original temperature for hours.