While the official airport website is silent on the topic, O'Hare airport guides indicate that animals are not technically allowed inside buildings aside from service animals (whose presence is permitted by law). For example, iFly states
Animals are not allowed inside the passenger terminals unless they are in a kennel. Exception: Working dogs on a leash.
Windy City Paws, a Chicago-based dog walking service, states in its airport guide
Working and service animals are allowed inside the passenger terminals as long as they are on a leash. All other animals must be in a kennel. Not all those traveling with pets adhere to these rules, and many can be seen walking their dogs on leashes throughout the airport. However, the rule is always in place and can be enforced at any time. The best route is to always have your dog in a kennel instead of taking the chance of missing your flight or being fined.
Anecdotally, I have seen leashed dogs which do not appear to be service animals waiting in the baggage claim area of the terminal, in line with the WCP FAQ, but I imagine if the airport personnel in your area are stricter, or if someone else complains, you would probably be asked to wait outside on the sidewalk. Also note that the pet relief areas are all located behind security, so you would need to take your dog outside somewhere when nature calls.
In general, an airport is not someplace I would casually bring a pet. They are noisy, crowded, full of strangers trying to pet or feed them, odd-smelling, and otherwise stressful to the animal. And an animal which is well-behaved at home may not react well to the craziness of the airport environment, which after all brings out the worst in humans.