The big difference between Uber and cabs, however, is that I left the driver a negative review on Uber, and their customer support got back to me. If he keeps pulling that crap he's not going to be able to drive for Uber anymore. There's essentially no recourse for cab drivers. Sure, they may have contact info posted for their company by law, but if you try to contact them you won't reach anyone who cares.
Diverging off-topic, and your account is light on specifics needed for this divergence...
So, sometimes it could be that the Uber (or taxi) driver knows more about the rules of the road that the passenger, and when the passenger complains based on wrong understanding of traffic law, it's not an easy customer-serving situation.
I'm pretty sure it's illegal to use a bike lane to cut off another car that's already stopped at an intersection. Keep in mind this is a road that has a single travel lane in each direction -- there isn't a separate right-turning lane.
Their association is also corrupt as hell, so they can't even claim that over Uber.