There are two styles of Thai iced tea: with or without milk. Tea with milk is called cha yen. Tea without milk is called cha dum yen.
You will see small coffee/tea shops sprinkled all over Thailand. However, unlike in the US or Europe, Thai people like their iced tea on the go and take it with them in a small plastic bag with a straw sticking out. Very few people sit at the shop and drink it leisurely. In fact, many shops have no place to sit.
Note: There seems to be a confusion about Thai tea. This recipe is Thai tea from Thailand, not Indian chai from India.