To any New Yorker, Uber is a black car service... you can't hail an Uber on the street. And the current compromise between Uber and NYC is that Uber cars are licensed exactly like black cars. Done.
(You actually refer to the street hail thing in one of your other comments, surprised you didn't mention that here.)
Just to straighten out the terminology: Let's say "Taxi" (at least for the context of this discussion) is a car that takes someone somewhere the passenger decides, for money.
It doesn't matter whether you hail it on the street or book it elsewhere. Taxi is a paid car. By that definition in NYC all black cars, yellow cabs, Ubers etc are Taxis.
That weird US regional laws have made a whole family of terms for this simple service just makes this discussion muddy (as you might have noticed).
What this news article is about and what is already obvious is that Uber is Taxi because they take people places for money.
Same in Germany, but Uber wanted to be licensed like a web service, not like a black car service, and argued that instead each driver is operating a black car service, and they should be responsible for insurance.
(You actually refer to the street hail thing in one of your other comments, surprised you didn't mention that here.)