Well, Uber is taking a heavy beating in Europe (and thanks to our regulations, taxi service never has been as bad as in the US, so the market for "better taxis" isn't as large to begin with), and AirBnB already bows to regulator's pressure all over the world.
Uber competitors didn't exist in Europe before Uber did. You are talking about actual "sharing economy" like the french carsharing service but thats a very different thing.
Airbnb didn't really exist either you are talking about things like Novosol that functioned as a middle man apartment renter.
1. Uber competitors existed – although those were usually financed by the Taxi companies.
2. AirBnB competitors existed in the way that you could rent your apartment to others via such services. Renting out private apartments was standard for decades and centuries even.
Taxi companies are regulated which means there was there is a limit to how many taxi drivers are allowed. Uber doesn't have such a limit and is thus undermining the market. Uber is planning on replacing all drivers with automated cars, how many taxi companies you think are working on that right now?
What you are talking about is car-sharing which is something Uber also does but not what they do mostly.
With regards to Airbnb. If airbnb ish companies existed before why didn't they then outcompete airbnb? Why is Airbnb fighting fights and not European rentals.
Anyway we are getting away from the main point which is that american companies dare take up a fight with legislation very few european companies if any do that. Because of reasons I mention in the essay. Tesla, LendingClub, KickStarter and so many others have had to find ways through the legislation system.
US companies just have a better culture for challenging incumbents and legislation. A much richer culture for lobbying and so on.