In terms of popular and well known local dishes, the deep dish pizza and Chicago dog are great. I agree with you there.
The one that people outside of Chicago don't know a lot about, that is still a delicious representation of the city, is Italian Beef.
And the one that is uniquely Chicago but isn't going to be winning over people in a blind tasting, is shots of Malort.
I've lived all over the U.S., so here are some of my favorites:
Texas:
Louisiana:
The American South in general:
Southern California:
New York:
Chicago: