I don’t think there’s any regulation on bringing your “normal” sized bike on buses, but may depend on if the driver is willing to take you and if you are willing to be on a bus load of people. Getting off may be a huge hassle if it is full of people.
Basically, as long as the bicycles are properly stored are treated as normal luggage like for example the folding bikes then they can be allowed on the bus
However Taipei has some buses that have a bicycle rack mounted on the front of the bus like 小18路線公車 and 台北客運藍46路線
Years ago a buddy and I wanted to ride the Maokong off-road trail but were too lazy to ride our heavy mountain bikes up the mountain, so we decided to flag down a 530 bus at NCCU. The driver was unwilling at first but still let us on after some hesitation (the bus was practically empty). We had a blast blasting down the mountain. That was the only time I ever took a bike on to a Taipei bus.