I'm pretty sure he didn't wear that till Conway, Estrada & Wood's revival of ALL-STAR. That being said: the costume you mean appeared before Earth-2 Robin wore it. In said story, he and the Earth-1 Robin, tired of being treated like a couple of kids by their respective elders, go off and try to do some stuff on their own in a dark cave.
No, not what you think.
Earth-2 Robin lends E1 this costume to wear so he can look more grown-up. It's his own. He says his friend Neal Adams designed it for him. Think of this: he HAS the better costume and still chooses to continue with this one, which is like the worst drunken cosplay thing ever. Meanwhile Dick-1 gets to wear it. One wonders why the fully adult Dick-2 would have a costume about that fits not his, but a teenager's proportions. The exact proportions of his counterpart. Hm.
The costume reflects one thing, certainly: Grayson-2, forever fretful as a young-'un that Batman-2 would replace him, apparently didn't stop being emotionally stunted and needy into adulthood. Besides this costume, there's his somewhat creepy attempt to "relate" to the other Robin and his deferential attitude toward the JSAers(which thankfully Conway later dropped)--Grayson, you're like, over 30. Grow some balls and find friends your own age.