German transport law separates two things that look identical from the pavement. A taxi may be hailed or taken from a rank, and it operates under a duty to carry, a duty to be available and — crucially — a fixed municipal tariff. It charges what the tariff says, metered, and it is not free to agree a price with you in advance.
A private hire car under § 49 of the Personenbeförderungsgesetz works the other way round. It may only accept journeys booked in advance at the operator’s place of business, it is not obliged to accept every job, and after each one it must return to base unless a further booking has already come in. In exchange it is not bound to the taxi tariff — which is precisely why it can agree a price with you before you travel and keep to it.
So the fixed price is not a marketing device layered on top of a taxi. It is a consequence of the licence the vehicle runs under. The operating company here is a registered hire-car business, and the pre-booking requirement is why this site sells journeys rather than a phone number to wave at a kerb.