You know what gets me. Robin was the name Dick's mom called him. The Robin colors are his family's colors. I like to think of Robin symbolizing a parents (particularly a mother's) love/protection that strengthened Dick. Now imagine Damian finding out that meaning and suddenly he realizes what Dick truly gave him, his family's love/protection. I imagine he'd be especially defensive/proud of Robin because Dick trusts him w/that privilege and how Dick wants him to feel a part of HIS family too.
Yes! A lot of people forget that Robin wasn’t just some sunny name Dick chose, it had special meaning to him personally. They don’t know that by taking that second name, he is reclaiming the love of his parents and using the goodness they taught him for good. It’s one of the reasons Dick was so incensed when Bruce nonchalantly handed over Dick's Robin costume/name because it wasn’t Bruce’s to give away. Bruce, like everyone else, had forgotten that to Dick it wasn’t just a title, it was a personal promise.
Once Dick gets to know Jay it’s cool of course but there’s this very symbolic scene right before Jay died where Dick hands over the original costume as a show of good faith that Jason has earned Robin and is worthy of Mary Grayson’s blessing. I’d like to think this little conversation/explanation happens to all the Robins as they understand just what they’re trying to live up to. It’s something that helps Dick bond with his successors and helps to keep the spirit of his mother alive in such a simple way. As long as there’s a Robin taking to the sky, her love will never die.
Damian takes this promise very seriously as he takes all his given titles. He’s very big on honoring legacies being worthy so I can imagine Damian trudging out to the cemetery where the Grayson’s are resting and quietly explaining that he is the newest Robin. He admits that he’s not like the other Robins, like Dick, but that he will do his very best to uphold the good name he has been blessed with and will do his best to honor her name. She raised a good child, one who could fly and light up the world with his smile, it’s now Damian job to see that such a tradition continues, even if it’s in his own unique way.