Am I really an intern or just an employee who isn't getting paid?

The answer is no. Calling a job an "internship" doesn't relieve an employer of the obligation to pay for your time. An employer that wants to pay less than the minimum wage (or nothing at all) to an intern must meet a strict six-part test, intended to ensure that the position really is a learning experience. If a job doesn't meet this test, the person holding it is entitled to be paid. 

Unfortunately, you've run into an increasingly common problem these days, as the job market continues to sputter along and new college graduates can't find entry-level work in their fields. Some employers have decided to take advantage of the huge glut of potential employees by creating unpaid positions to get their work done free. Especially in fields that are competitive and glamorous (like yours), employers have been mostly getting away with this cheat. 

But that doesn't make it legal. Interns are supposed to be temporary workers, typically students or people who are new to the field, who take a position to learn what a job is like and get some experience. If an employer doesn't pay an intern at least the minimum wage for each hour worked, the internship must meet a six-factor test created by the federal Department of Labor. The Department looks at the factors listed below to decide whether an internship is really a benefit to the intern or is actually benefiting the employer:

• The internship must be similar to training that would be provided in an educational environment. An internship that centers on academic experience is more likely to meet this requirement, as is one that trains the intern for multiple employment settings rather than just to do the work of the employee. 
• The internship must be for the benefit of the intern. The more the business depends on the work of the intern, and the more time the intern spends on routine, regular work of the business, the more likely the intern will be considered an employee who is entitled to minimum wage -- and overtime. 
• The intern must not displace regular employees and must work under close supervision of existing staff. A company that is using interns rather than regular employees (either because it has let employees go or put off hiring) is likely to be seen as an employer that owes wages and hour obligations. Similarly, if an intern receives the same supervision as regular employers, that argues against a true internship experience. 
• The employer gets no immediate advantage from the intern's work -- and may, on occasion, find its operations impeded by the internship. Having to provide an intern with extra training, special learning opportunities, and additional supervision and guidance, for example, will entail time and expense taken away from other activities, and is less likely to indicate an employment relationship. 
• The intern is not necessarily entitled to a job once the internship ends. An internship that is actually a trial period, the successful completion of which will lead to a job, will be considered regular employment that should be paid. 
• Both the employer and the intern understand that the intern is not entitled to wages for time spent in the internship. If the intern thought the position would be paid, that argues in favor of employment status. 

As you've probably noticed, your "internship" meets exactly none of the requirements on this list. You are receiving no training, supervision, or benefit from your work. Not only are you doing entry-level office work, but you are also being pushed into service as a baby-sitter, dog walker, and delivery person for your employers. If you were attending meetings with clients, going along on shopping excursions for fabric and materials, observing the design process, learning how to fit models, and otherwise learning your trade, your employer could ask you to fetch the occasional cup of coffee or copy marketing materials without jeopardizing your internship. However, your employer is providing no learning experiences, and instead is requiring you to do work it would otherwise have to pay for. This is illegal.