Getting a taxi concession was no joke. You literally had to study on the layout of the city. And given that those are European cities, there is no grid system. You also had to know about safety and take first aid courses.
Additionally, taxis require special inspection on top of the rigid inspections necessary for cars in Germany (you wouldn’t want to endanger people with a rust bucket on the autobahn). So yes, they were exceptionally safe.
Taxis are mostly Mercedes in Germany, as traditionally Mercedes had the most experience with taxis, and the proper programs in place for taxi use. When I see a Mercedes E wagon on a freeway, I still think "taxi" (like some think police car when they see a Crown Victoria).
Source: There was a time where I took a taxi almost every day in Germany, and a close family member owns a German taxi concession for many decades now.
Lyft+Uber are an absolute joke in comparison.
(I talk mostly in the past tense, because I haven’t lived in Germany for long now. I don’t know whether the gig economy damaged the enormous quality of taxis over there.)