Hmm. The hot ginger drink I've usually had is called "susu jahe", but maybe there are other popular ginger drinks in Indonesia that I am unaware of. Also, I translated this sentence originally to "I like hot ginger drink", because you are right, "I like drinking hot ginger" is not a usual sentence in English. But why should it be? ¯_(ツ)_/¯
It’s kinda similar like hot chocolate or a cup of coffee. The fruits have the similar name, it doesn’t mean that we have to add ‘tea’ or ‘drinks’ just because it’s diluted with water. Especially in context of jahe as drinks in Indonesia, jahe panas and the jahe have different level of concentration.
In the Indonesian context, people may drink "jahe panas", but in the English language, you are implying that you warm up the ginger piece directly (without any liquid) and eating it just like that, which doesn't make sense. That's why we say "ginger tea" because we infuse the hot water with the ginger, rendering the liquid to be "hot ginger tea."
When I was in Indonesia and saw Ginger on the beverage section, they will always add teh on the end or put it in the teh section.