It is usually OK to screw through those holes when mounting a PCB in a regular appliance. A bike is going to be a bit tougher than that.
- Don't apply too much torque. Screw holes which are surrounded by a ring of vias can take more torque than one which is just a plated through hole with a big pad.
- Think about where you are screwing it on the bike frame. You might be surprised how much a bike frame flexes when it's being ridden. If the screws move even 1mm relative to each other, it's going to crack the PCB quite fast.
- Think about vibration. If the bike is being ridden outside of a velodrome, the vibration could damage the solder joints on larger/heavier parts. Rubber grommets or standoffs would transmit less vibration to the board than if you just screw it on.
- Think about dirt and road grime, especially road salt. You might want to enclose the PCB, in which case you'd be better off bolting the PCB to the inside of the enclosure, and the enclosure to the bike.
If it was me, I'd probably fix the PCB inside a diecast box with rubber/flexible mounts, then bolt that to the bike.
Also, your question was about mounting so I've only discussed mounting, but the vibration thing is a big potential problem. Mounting with rubber mounts will reduce the vibration reaching the board, but depending on the speed/surface the bike is being ridden at/on and the components used on the PCB it may not be enough. Also consider component selection, securing large components with glue, encapsulating or potting the board, or just living with high failure rates.