It's hard to translate. The law speaks of Erholungsurlaub, which could be translated as “recuperation vacation“. That already sets a clear purpose. There is also a clause that prevents us from working during vacation if that work endangers the purpose of the vacation (that is, recuperation): https://www.gesetze-im-internet.de/burlg/__8.html

Other countries have non-competes, employer ownership of things created during off-hours, and so on.