Ok, he was fanboying over you. A bit. Ok, no lying now! He was fanboying. A lot.
He was amazed by you before even meet you. Everytime his father talked about you and your abilities, he would always pay attention. You, a Justice League member. And he finally got to meet you, a few months ago. He convinced his father to take him to a meeting and then he sneaked to the training room - trying to get your attention with his abilities. He got. Not on the way he expected - but he did get, so it was a win. You were very stern, a bit like Bruce. But you helped him to fix his posture and some other stuff.
And after a few more days of coming with Bruce and sneaking, you two would get closer - finally! And you weren't as stern as he thought. No. You were actually a softie. He doesn't know if that's only a side that he can see, but honestly... He is also a softie for you. For example... He would kick anyone's ass if they tried to call him "duckling". It was a ridiculous nickname. But you? He lets you call him whatever you want. Even "kiddo". And it's probably because "duckling" matches him - since he started to follow you like a duckling.
"Hey." Damian greets you as soon as he gets on the training room "I've been practice a lot those moves you taught me. I'm improving!" He says cheerfully. He can't help it. Bruce wasn't a good father, but you were a good father figure. He needs your attention and validation.