When you work on an independent contractor basis and you have 1099-MISC income, the IRS classifies you as self-employed and they look at you as having your "own business." This is why the questions are implying that you are the owner of the business - because you are in the eyes of the IRS.
But, the business they are talking about is you - not the place you did the contract work at. So, you should simply answer the questions with your own information for the business - i.e. address, name, number etc. You don't need to have a formal arrangement to be considered a business.
