Heat a Dutch oven or large skillet over medium-high heat. Cook and stir sausage in the Dutch oven until browned and crumbly, 5 to 7 minutes; drain and discard grease. Add onion and garlic; cook and stir until onion is tender, 5 to 10 minutes.

Stir chicken broth, tomatoes, carrots, celery, ketchup, and red wine into sausage mixture; bring to a boil. Reduce heat, cover, and simmer until carrots and celery are tender, about 30 minutes. Add tortellini and zucchini to the soup and cover; simmer until zucchini are tender, 25 to 30 more minutes. Ladle soup into serving bowls and top with Parmesan cheese.