Yeah, Annyong isn't really much of a character in and of himself, and I kind of doubt he was meant to be. I think he does a good job of just showcasing Lucille's rash and blatant disregard for anyone but herself. He was only adopted because Buster wouldn't finish his cheese, she kept him because he was making Buster jealous, she hid Nana's money in an account for him, and she used him as a purse. He definitely does a good job if his purpose is to show what a horrible (in a hilarious way of course) person Lucille is.
"Hey, who called the cops??"