Can a cyclometer be used with any bicycle?
The cyclometers I have owned counted wheelrevolutions by impulse of a circular lever on the cyclometer itself, most often mounted on the front fork. The impulse was from a solid peg mounted to a wheel spoke.

I have not seen such a cyclometer in the past 25 years.  Bike computers perform the similar functions, as well as others, depending on specific models.  Most bike computers operate from magnetic sensors which can be mounted or adapted to almost any cycle. Some feed the data from the sensor to a head unit mounted were it can be seen by a wire. In a few cases the supplied wire is too short to reach from a sensor (generally for pedal revolutions) to the head unit or alternative smart phone or watch read out unit.