As a German horse riding seems to be to the point. Why is it horseback riding in English? Isn't it obvious that you ride on the back of the horse? Is there a difference between British and American English on this point?
It's a good question. As TomH wrote, horseback riding is a primarily American usage. It's not in English generally.
I have not heard anyone in the UK use it. The only time that I heard it was in American films. I remember it because to me as a British person, it sounds odd and unnecessarily long. I remember thinking to myself, “Why say horseback riding? What other part of a horse do people ride on?”
In the UK, it is obvious that riding a horse means riding on its back.
Use of the word back in this is just an unnecessary extra. Whenever it is talked about here in the UK, people just say horse riding. It is also enough to say riding, because it is obvious what is meant by the context.
The difference between the two ways of saying it depends on the regional flavor of English being used. Horseback riding is used by people speaking American English, and horse riding is used by people who are not speaking American-style English, including English and other British people.
It is interesting to compare the use in British and American English. Here is the NGram for the British English corpus:
Now look at the American English corpus:
In British English the two expressions were roughly equally frequent until 20 or 30 years ago, when the "back" variant rapidly became four times more frequent. In American English it seems to have been consistently four times more frequent. Perhaps this is an indication of the American usage being assimiliated into British English.
So, the answer to "is it British English" seems to be: no, not until recently.
The first horse races I went to, at a county fair back in the 50s, did not have the jockeys sitting on the horses' backs, but rather in sulkies drawn behind the horses. Trotting races. And from the following lines in the Music Man, after comparing the corrupting influence of pool (as compared to wholesome billiards), Harold Hill sings in Ya Got Trouble,
And list'nin to some big out-a-town Jasper Hearin' him tell about horse-race gamblin'. Not a wholesome trottin' race, no! But a race where they set down right on the horse! Like to see some stuck-up jockey'boy Sittin' on Dan Patch? Make your blood boil? Well, I should say.
I gather there was once considerable feeling against horseback racing. Hence the need to distinguish it, terminologically.
(The song is here: https://www.youtube.com/watch?v=LI_Oe-jtgdI )
Because it was so commonplace before the motor car arrived, you just said "riding" and so do all people today who regularly ride, or who have grown up riding horses. You talk about "going out for a ride" .That is why "do you ride" is usual. If cycling is the alternative, you can say "I'm going out on my bike" or "I'm going for a ride on my bike". Motorcyclists are often called "bikers" so that covers them!
All this horseback stuff is maddening - it is just RIDING!