I would like to verify the claims a small business is making, regarding ownership.
How can I research this and see if they are making false claims, or misrepresenting the real nature of their relationship?
The business is located in the USA, however if you have relevant information on other countries, I would be interested in that as well since I foresee myself doing lots of this research in the future
Check with the Secretary of State for the state in which the business is incorporated. You'll be able to view or request copies of various public records about the business. Depending on the state and the legal structure of the business, these records include ownership information.
(It's usually the SoS that handles incorporation records, but is sometimes a separate division of the state government like the Division of Corporations. Googling for corporations should get you to the right place.)
Try Internet searches. Unfortunately might be tough to get ownership information for a non-public Delaware company, for example, although other Secretaries of State might have more info.