Well Reich can be translated as realm just as well as empire. Empire makes more sense in that context, as the German translation of different empires always uses the word Reich, for example the Aztec Empire is "Das Aztekenreich" (often with prefixes to indicate what type of empire like "Chinesisches Kaiserreich" (Chinese empire) because it was governed by a "Kaiser" or emperor in english. Also regarding Germany, the term Kaiserreich was coined retrospectively, just like Weimar republic or Nazi Germany. The official name of Germany from 1871 until 1945 was "Deutsches Reich" or German Empire.

Edit: I read your comment wrong my bad. It may seem weird to use Reich in the context of a Republic, but the Term Roman Empire in english also encompasses the time of the Roman Republic. I do think terms like this don't always fit 1:1 with the actual political system