However, you are right in that there could be intellectual property issues (namely, trademark) when two companies have the same name. If you have any thoughts or plans for branding your business on a nationwide (and not just local) level, you'll want to be aware of anyone else using the same or similar name.
Depending on the nature of their business and the trademarking steps they have taken, this other company could become an obstacle to you in attaining trademark protection for your company.
As you are already thinking along these lines, you may want to consult with an intellectual property attorney so that you can develop a sound strategy for how to proceed.
Nina L. Kaufman, Esq. is an award-winning New York City attorney, edutainer and author. Under her Ask The Business Lawyer brand, she reaches thousands of entrepreneurs and small business owners with her legal services, professional speaking, information products, and LexAppeal weekly ezine. She also writes the Making It Legal blog.