Sanskrit is similar. There are elaborate rules for combining, called "Sandhi" [1] [2] which shows how nearby vowels or vowels near consonants change to form a combined sound "aa+e = aie" etc. This allows the formation of fairly long words, for example - "kimkarthavyathaavimoodatha", "not knowing what to do next".
Sanskrit has analogous structures, but they are not called Sandhi. Sandhi refers specifically to phonetic combinations of adjacent sounds in subsequent words, not a semantic or morphological combination. These phonetic processes are almost completely independent of semantics.
The analogous structures in Sanskrit to the articles German examples are various types of compounds, like tatpurusha, dvandva, etc.
[1] https://en.wikipedia.org/wiki/Sandhi
[2] https://ubcsanskrit.ca/lesson3/sandhirules.html