While it is really hard to prove a negative, I would say that no, this exact dish seems to only have arisen once, and is only common among East Slavic people - which is a better way to describe its origin area than by modern country borders.
Starch-based soups do exist around the world as a poor people food, although they are not as popular as various types of gruel (which is the cheaper preparation, since milling is an expensive process in low-energy societies). But on most continents, wheat is not the dominant cereal. So for example in Japan, people traditionally eat a gen-mai soup (rice based), and similar is true for other continents. So it should be sufficient to look at the most similar dishes in Europe and see if they are virtually the same.
European countries do have traditions which are highly similar to this soup, but they happen to have developed slightly different in their detail. None seems to combine all the central features of this specific dish.
- Having a soup made from barely-shaped dough. This positions the zatyrka in the middle between a noodle soup, a flour soup and maybe a bread soup. In noodle soups, the noodles tend to be elaborately formed, for example in a ravioli soup. Also, when talking of "noodle soup", most people imagine a soup that is broth-based and has vegetables added. Standard flour soups don't have any shaped lumps at all, and are anyway rare outside of the same Eastern Slavic culinary region, except for a few exots like the Basel flour soup, which is more of a drinkable gravy. The closest geographically separated relative of the zatyrka in this sense is probably the trahana, which is made in ex-ottoman regions. There, sourdough is cooked into a soup, but without shaping, just dissolving it in water. The zatyrka process is halfway between these two ideas, and doesn't seem to be practiced in other regions.
- making noodles by rubbing. Examples of this exist, e.g. Schnupfnudeln and Spätzle, but they are not the same process, don't produce the same shape, and aren't traditionally made into a soup.
- making a soup with a milk base. This is quite common in dairy producing areas. The less similar examples are chowder-like, but you can find dishes like Kappeler Milchsuppe (Switzerland) spread throughout the European continent, which involve cooking up bread, grains or noodles in milk. There are also sweet variations.
I must admit that I am not very familiar with the cuisines of the countries around the Caspian sea, which would be a good candidate to have similar foods, maybe they have similar dishes which I have missed.