It appears that the product closest to American cultured buttermilk is Dickmilch. As noted here,
This fermented dairy product known as cultured buttermilk is produced from cow's milk and has a characteristically sour taste caused by lactic acid bacteria. This variant is made using one of two species of bacteria—either Streptococcus lactis or Lactobacillus bulgaricus, which creates more tartness.
From the German article on Dickmilch,
Im Unterschied zu Joghurt (thermophile Kulturen, Temperaturoptimum 42-45 °C) werden bei der Herstellung von Dickmilch mesophile (Temperaturoptimum 22-28 °C) Streptokokken-Kulturen beigefügt (Streptococcus lactis, bzw. S. cremoris statt S. thermophilus). Der Milchansatz wird anschließend bei Temperaturen von ca. 25–28 °C über 15–20 Stunden dickgelegt.
In sum, yogurt is produced with thermophilic (those that like high temperature) bacteria, while Dickmilch is produced with mesophilic (medium temperature) bacteria, specifically lactic bacteria in the Streptococcus family, which is basically the same as mentioned in the American buttermilk article above.
The description of the production method of Dickmilch -- 15-20 hours at slightly warmer than room temperature -- accords with the standard method of making American cultured buttermilk. Here, for example a standard recipe recommends 24 hours at room temperature, but I've also seen recommendations for a "warm" (though not hot) spot for 18 hours or so when making buttermilk at home. The bacteria are happy over a relatively wide range in my experience making cultured buttermilk, and the timings will vary depending on temperature and strength of the culture.
The one difference you may encounter with the German Dickmilch compared to American buttermilk (assuming these quotations are accurate) is that American buttermilk can vary in its fat content. This is somewhat of a regional phenomenon. In most regions of the U.S., cultured buttermilk is produced from low-fat (~1%) milk (or sometimes even lower fat). As far as I know, this is done to approximate the characteristics of traditional buttermilk produced from butter churning of soured cream (not "sour cream" -- cream that has "matured" from fresh milk), which would usually have a very low fat content, since almost all the fat would glob together in the butter. (As an aside, I have substituted traditional buttermilk -- from butter production -- successfully for cultured buttermilk in some recipes, but you need to start with soured/cultured cream. So you might be able to use Buttermilch as well... though it's probably not what your American recipes assume.)
In recent decades, since Americans have basically forgotten the taste of traditional buttermilk (and many have even forgotten butter produced from soured/cultured cream), the idea of cultured "buttermilk" has become rather abstract. Also, the standards for dairy product labeling allows cultured buttermilk to be labeled "low-fat," which makes sense since it's made from low-fat milk, but also doesn't make sense in that the fat content approximates the fat content of traditional buttermilk.
This seems to have led some American buttermilk producers to tend toward lines of "full-fat" cultured buttermilk, which is typically found in the Southeast U.S. It is a bit of a ridiculous label, since it strays even further from the original product that cultured buttermilk is imitating, but there it is. This "full-fat" version is generally made from whole homogenized milk. (Note that I have no problem with people making the stuff, but calling it "full-fat" -- often authentic "buttermilk" is just weird.)
I'm assuming that the German Dickmilch is probably produced from regular whole milk (the article I cited above refers simply to "homogenized cow's milk," presumably with some fat content), so your Dickmilch might be closest to a Southern U.S. version of buttermilk. Most American recipes don't specify the fat content of the buttermilk, but they probably assume the more widely-available 0.5-1% or so fat version. For most recipes, this won't matter, but if it does, you could adjust the fat content of other ingredients slightly.
(Also, by the way, the only brand of traditional buttermilk produced from butter that I know of sold commercially in the U.S. is made by Kate's of Maine.)