The episode seems to be based around the events in the past that led to the breakup of the duo, after which Dick went on to be the resident, and lauded, superhero of Bludhaven (perhaps closest to the marvellous "Duo Dynamic" take on Dick's superhero career from Gotham Adventures #12), so he hasn't been Batman's partner in some years and HAS made a name for himself as Robin, but hasn't felt the need to shed the name "Robin", since it's the hero identity HE created, and there is no Jason Todd waiting to replace him.