I suspect your gut bacteria are simply unused to the new environment (possibly including the milk). The same kind of thing happens to most people when they travel overseas. As a child I spent most summers in Mexico. Within the first few weeks I would have a bout of serious diarrhea and vomiting, literally every year. But native Mexicans who ate the same things didn’t have the same problem, being well adapted.
Anyway, what specific type of milk are we talking about? Full fat homogenized milk? Skim milk? Unhomogenized milk?
In the German case you might also be talking about UHT milk, which is common in many countries (not sure about Germany) but vanishingly rare in the USA? I think that stuff is awful, YMMV.
There are many different kinds of 'American milk'. A primary difference is what the different farmers feed their cows.
A few years ago I suffered through the History Channel's Modern Marvels episode about cotton. There's a bit towards the about how cottonseed meal (a waste product of the cotton industry) is commonly used as feed for the dairy industry.
Someone commented about how they taste a difference for the week after farmers switch their cows from summer rations to silage -- it takes a week for the cows' bacteria to adjust. I should find that comment & favorite it... Ah, here:
You can always tell when the cows switch
between grass and silage in the spring
and fall. Milk tastes like garbage for a
week or two until they get sorted out.
The usual macronutrient values (9kcal/g fat, 5 kcal/g EtOH, 4kcal/g carbs and protein) are surprisingly decently calibrated for metabolic inefficiencies.
They do not, however, take into account hormonal responses, which are probably quite a bit more important than once assumed. So there's that.