I'm not yet at the point of wanting to license bicycles, but (if I should get that crotchety some day) I hit on a better plan.
The main area where bicycles conflict with cars, trucks, trains, and buses is downtown Portland. Just as New York (so I'm told) imposes a higher age limit on drivers in Manhattan than in the rest of the state, we could have an enhanced requirement for bicycling in downtown Portland. I define downtown Portland, for this purpose, as the area between Front Avenue and Interstate 405, starting at Market Street and going north to Burnside Street. This doesn't affect Waterfront Park and Portland State University.
Instead of licensing bicycles, let's license bicyclists. Make it unlawful for anyone to bicycle within downtown Portland unless he or she is wearing one of those bright yellow bike jackets. The wrinkle is that the City would sell the jackets, and each jacket would have the cyclist's license number prominently displayed on the back, rather like a license plate printed on Gore-Tex. No cyclist could get a numbered jacket without first attending a traffic safety course. Each cyclist would be responsible for violations of traffic laws committed by whoever is wearing that cyclist's jacket, just as the owners of cars are responsible for their parking tickets regardless of who parked the car.