It's remarkable how many men (sorry to be sexist, but in my experience it is usually men) try to turn their spouse/girlfriend into a female version of themselves. They like cycling, they try to push her into liking it too. They like golf, they buy her some lessons she doesn't want.

Maybe the only reason she says she wants to ride her bike is that she is trying to please you. It certainly sounds as if you might be forceful enough to elicit that response. I suggest you back off, leave the bikes where they are and enjoy the fact that you and she have different and separate interests. God knows it would be terrible to have to do everything together...