The results are not conclusive.
There are two main studies that regard that Issue. The study Chicken Soup Inhibits Neutrophil Chemotaxis In Vitro, suggest that:
The present study, therefore, suggests that chicken soup may contain a number of substances with beneficial medicinal activity. A mild anti-inflammatory effect could be one mechanism by which the soup could result in the mitigation of symptomatic upper respiratory tract infections.
And the study Effects of drinking hot water, cold water, and chicken soup on nasal mucus velocity and nasal airflow resistance compares chicken soup to hot water and concludes that chicken soup is better that hot water:
We conclude that drinking hot fluids transiently increases nasal mucus velocity in part or totally through the nasal inhalation of water vapor. Hot chicken soup, either through the aroma sensed at the posterior nares or through a mechanism related to taste, appears to possess an additional substance for increasing nasal mucus velocity. Finally, hot liquid might be superior to cold liquids in the management of fluids in upper respiratory tract infections.
However, as this NYT article points out:
None of the research is conclusive, and it’s not known whether the changes measured in the laboratory really have a meaningful effect on people with cold symptoms
One additional factor that should be taken in, is that the symptoms of flu and the common cold include fatigue and loss of appetite, which may cause a sick person to not want to eat any food and by thus preventing the intake of important nutrients. Chicken soup, which is made primarily from chicken and vegetables, contains many nutritious substances that can be digested through drinking and by that help the recovery process. As the NYT article states:
However, at the very least, chicken soup with vegetables contains lots of healthy nutrients, increases hydration and tastes good, too.