To add to Ambicion's and Maarten Buis' excellent answers, here are my two-cents based on getting a doctorate in Mathematics in Germany and then teaching in the US. From what I can tell, things in the US have not changed enough in Mathematics (I now teach in CS).
Beginning Mathematics in Germany was more abstract and used the axiomatic method, whereas for example Calculus in the US is aimed at natural scientists and especially engineers (at least at my school, the engineering school drives what is to be covered in Calculus). The equivalent class in Germany is Analysis. In the US, students are taught to calculate integrals and derivatives so that they can later do Fourier analysis and similar transforms, even though most of the students do never use them. In Germany, we were taught how to derive analysis from first principles and how to prove the important theorems. In the US, the quotient rule of differentiation is not often proven, whereas it is standard in Germany. In the US, students learn how to prove theorems maybe in the last semester of their sophomore (second) year and usually later. In my school, this is done with Discrete Mathematics and with Linear Algebra.
From what I can see what is happening in Germany right now, the engineering schools will teach their own Ingenieurmathematik classes as some did in my years, just after the Romans left the Rhineland.
So, even with the change in orientation towards the new B.S. degrees, I would still assume that beginners classes in Mathematics have a different goal than beginners classes in the US.
I am just supporting here what Maarten Buis and Ambicion wrote, even if I am not repeating their arguments. There are also great observations in the comments. This is a very interesting question that you are posing, worthy of a much deeper investigation.