It's not so much that I think he sees her as a damsel (clearly she isn't!) as that it's kind the path I think his thoughts might take based on the situation. I mean she's just sort of missing, so he's bound to wonder about where she is and thinking she's probably in trouble. And obviously he'd be feeling a lot like Sokka felt at the beginning of the Boiling Rock, except nobody will answer his questions or give him a way to actually do something. So I could just see it as something he might think about when he was younger in order to not think about the worse things that might have happened to her and keep his hope alive. If she's being held captive, she's not dead, etc.

Though I could also see him wanting to protect her because she protected him.