What type of certification, if any, should I expect my interior designer to have?

In New Jersey, the state regulates designers so check with your state. This is what NJ requires:
A New Jersey Certified Interior Designer, (NJ-CID #000), signifies an individual who has met the state’s requirements for interior design (Experience, Education and Examination), and holds a valid certificate issued by the New Jersey Interior Design Examination and Evaluation Committee, under the State Division of Consumer Affairs.