The real Gustav Hartmann was not categorically opposed to motor cars and their use as taxis. In fact, he himself did own a motor taxi.
The horse with which the real Gustav Hartmann completed the journey of more than 1000 km from Berlin to Paris and back in 1928, was a gelding called Grasmus.
The film takes place in 1928.
The real reason for Gustav Hartmann to go from Berlin to Paris was his meeting with a French woman named Rachel Dorange at the train station in 1927. She told him that she is on the way from Paris over Berlin to Bucharest by horse. He promised her that next year he will visit her in Paris and will do the same by carriage. With his trip he also wanted to support the German-French friendship, especially after WW1.
When talking to himself in bed Hartmann mentioned Dr. Adenauer. Dr. Konrad Adenauer was the mayor of Cologne at the time of the movie's setting (1928) and the federal chancellor of Germany when the movie was made (1958).