These days we sometimes refer to a person's fixation as a hobby horse. Or, of course, a "hobby" - now the common English term for a preferred pasttime on which a person spends an inordinate amount of time.
As I mentioned in another blog, a "hobby" was originally a kind of riding horse. Hobby horse then came to mean a horse in a mummers' play - the original pantomime horse, although the costumes were more wicker than cloth.
And then? Well, let's see. Hobby horse has come to mean what Americans call a stick horse, has been used to refer to carousel horses.
In other words, a hobby horse became a "fake" horse...and one might say that riding a fake horse is a waste of time. Thus, a hobby. (I disagree that they're a waste of time, but...)