Similar to in Hungary. In hungarian milk bar is called "tejivó". There is a popular chain nowadays running under the name. This is what they say on their website about the history of the name:
"The milk bar ("tejivó" in Hungarian) is no new invention. Its history in Budapest dates back to the early 20th century. Such shops used to sell beverages made with milk, soft drinks and pastry. Some Hungarians still recall the exceptional atmosphere of these establishments, and the splendid feeling of sipping a steaming cup of coffee while eating a delicious croissant."
"The milk bar ("tejivó" in Hungarian) is no new invention. Its history in Budapest dates back to the early 20th century. Such shops used to sell beverages made with milk, soft drinks and pastry. Some Hungarians still recall the exceptional atmosphere of these establishments, and the splendid feeling of sipping a steaming cup of coffee while eating a delicious croissant."
https://www.cserpestejivo.hu/why-the-name-tejivo.php