Is there not a similarity between a bicycle saddle and a horse saddle that necessitates prohibiting women from riding bicycles, in analogy to the prohibition of riding a saddle which I read about previously?

It is not permissible for a woman to ride a bicycle in front of non-mahram (definition) men, because it often reveals part of her 'awrah (intimate parts), or her clothes describe her body, or she may fall and her 'awrah becomes exposed. As for the hadith (definition), "May Allah curse the vulvas on the saddles," it is false and has no basis.