IANAL but a bit of googling revealed that apparently German taxis are subject to at least these laws or parts of them: (I'll selectively paraphrase; there's quite a bite more in there.)
Contains rules for passenger transportation with trams, trolleybuses and motor vehicles. Apparently trains are covered elsewhere. Only some of the rules apply because cars (vehicles that can transport up to six people including the driver) have a special exception.
Verordnung über den Betrieb von Kraftfahrunternehmen im Personenverkehr (BOKraft):
Contains rules for passenger transportation companies that use trolleybuses or motor vehicles. This seems the most important one for taxi companies and covers things like vehicle maintenance, whether subcontracting is allowed, notification requirements, how to deal with lost property
The taxi-specific sections cover things like technical requirements, such as an alarm wired to the horn and lights that the driver can activate from their seat, a calibrated and illuminated taximeter or an optional bulletproof divider. Taxis must be painted with the color RAL 1015 of the RAL 840 HR palette and must have a "TAXI" sign of specific orientation and dimensions on top. They must display their taxi registration number in a specific style and place and also display the name and address of the company where the passenger can easily read them. Taxi drivers must take the shortest possible route to their target; if another route would be cheaper or faster, this has to be cleared with the passenger beforehand.
There's also some stuff in there that most people don't know - for instance, BOKraft-covered transport vehicles must have a copy of the laws governing pricing pnboard and must show them to the passenger upon request.
Berufszugangsverordnung für den Straßenpersonenverkehr (PBZugV):
Contains rules on who is allowed to transport other people. People with a criminal record or a record of severe traffic law violations are banned from working as drivers; company-level misbehavior might disqualify an entire company. Companies must have enough money to keep their fleet in shape. They must regularly check whether all drivers are still qualified to work as taxi drivers.
Drivers (in order to be hirable) must have an understanding of the laws governing passenger transportation, of vehicle maintenance, of radio protocols, of certain accounting procedures and even of environmental guidelines on vehicle operation and maintenance. They must pass two written and optionaly one additional oral exam of one hour each with the local chamber of industry and commerce; alternatively, five years of work in a different BPZugV-covered company can be seen as equivalent.
Paragraph 48 Fahrerlaubnisverordnung (FeV):
contains rules on taxi driver licenses. Examples: Taxi drivers must prove they know the area they operate in and that they have an appropriate understanding of first aid. If the driver is found unreliable, the license can be revoked (e.g. this once happened after a driver repeatedly refused to make short distance trips). Taxi driver licenses have to be reapplied for every five years.
Others, like the FPersG and FPersV, cover legal technicalities like when and how to have your license card with you etc. Additionally, municipalities may pass additional regulations.
So yeah, the law is pretty clear: None of the people who work for Uber are licensed to do so, thus they can't guarantee that they know about stuff like applying laws or where to drive. They can't even guarantee that the drivers aren't explicitly banned from working as drivers. Of course the law is going to come down hard on them.
If ridesharing is here to stay the law might adapt, but only by relaxing the signage requirements for very small companies. You'd still have to have a taxi driver's license, you'd still have to register the car and you'd still have to demonstrate an understanding of everything in appendix 3, PBZugV
. There's no chance they'll let "But, the internet!" trump regulations that, to me, are either sensible stuff to guarantee an acceptable level of service or bureaucratic overhead to make the sensible stuff work.