Islam cares about the relation between people. Good relations are encouraged. All in all, ethics and morals should be applied. Close friendship between girls and boys are un-Islamic. Islam does not allow this form of friendship for it mostly leads to illegal practices. Likewise, Islam does not allow physical touch or being in close contact between boys and girls, for this incites desires and opens the door to temptation.
Given the above, a boy should not ride a bike with a girl. This is not acceptable in Islam. Islam cares about good and ethical atmosphere and relations between people.