"Kakao" in German means cocoa, either bean, powder or hot cocoa. It does not mean the same as hot chocolate or "Heiße Schokolade." It can be drunk cold or hot. When it is made from powder mixes it is often called "Trinkschokolade" (popular in Germany are Kaba, Nesquik, Kakaoexpress) and is often served to children.
"Heiße Schokolade" (hot chocolate) on the other hand, is not made from cocoa powder or mixes. It is only called hot chocolate when real chocolate (like a chocolate bar) is used. It can be melted and stirred into milk or water, it often has added sweetener and a "Haube" or cap of whipped cream.
In Spain, hot chocolate is practically a national drink. The Spanish style is dark brown and thick, made from milk and bitter chocolate, thickened with a bit of starch (corn or potato). Recipe here In Mexico and South America, instead of the purified starch they use cornmeal to thicken their hot chocolate. Recipe here
In Italy, the hot chocolate is thickened with a high portion of bitter chocolate and extra cocoa powder. Recipe here
In France, bitter chocolate is used in their beloved, hot drinks. It is also made by melting the chocolate in hot water and then mixing with milk. In Café Angelina in Paris, the Chocolat l'africain (see one minute video) is whisked in a water bath until creamy, a rather time-consuming process. It is served in a small pot, with a cup and saucer, a bowl with unsweetened whipped cream and a glass of water. Recipe here
In Britain and the US, hot chocolate was not as popular as on the continent until the last hundred years or so, when instant cocoa was created. When it is made, it contains a high percentage of milk chocolate or cocoa and milk, is rather light, and in the US it is served by placing marshmallows on top of the hot drink. Instructions here
Hot Chocolate from Vienna, Austria ("Heiße Schokolade Wiener Art") is thickened with an egg yolk. This is often the way it is done is Switzerland too, with an extra sprinkle of salt.
When Germans make hot chocolate or Schokomilch, it is usually for children and usually from powdered mixes. There is a trend to real hot chocolate made with bitter chocolate, most often in the French style.