My business name is a homonym of another business name. Is this legal?

As my colleagues have correctly stated, the short answer to your question is NO, it may very well not be "legal" and expose you to liability for trademark infringement. The test for whether or not a mark infringes upon another's allegedly similar mark is whether or not the similarities of the marks and goods/services create a likelihood of confusion among consumers as to the source of each party's goods/services. The longer, more thorough answer is that the complete test for whether or not a mark infringes upon another is the following dozen or so factor test from some famous trademark infringement cases (the factors and tests vary somewhat among different states and federal jurisdictions because of different precedential cases but are conceptually the same): (1) The similarity or dissimilarity of the marks in their entireties as to appearance, sound, connotation and commercial impression. (2) The similarity or dissimilarity and nature of the goods or services as described in an application or registration or in connection with which a prior mark is in use. (3) The similarity or dissimilarity of established, likely-to-continue trade channels. (4) The conditions under which and buyers to whom sales are made, i.e. “impulse” vs. careful, sophisticated purchasing. (5) The fame of the prior mark (sales, advertising, length of use). (6) The number and nature of similar marks in use on similar goods. (7) The nature and extent of any actual confusion. (8) The length of time during and conditions under which there has been concurrent use without evidence of actual confusion. (9) The variety of goods on which a mark is or is not used (house mark, “family” mark, product mark). (10) The market interface between applicant and the owner of a prior mark: (a) a mere “consent” to register or use. (b) agreement provisions designed to preclude confusion, i.e. limitations on continued use of the marks by each party. (c) assignment of mark, application, registration and good will of the related business. (d) laches and estoppel attributable to owner of prior mark and indicative of lack of confusion. (11) The extent to which applicant has a right to exclude others from use of its mark on its goods. (12) The extent of potential confusion, i.e., whether de minimis or substantial. (13) Any other established fact probative of the effect of use. You need to consult a trademark attorney with your specific marks and situation to properly determine the extent of your infringement liability and it is strongly advised that you consult such an attorney to at least search whatever mark you want to adopt and clear it so it does not pose a risk of infringing any other marks in any relevant business. Beyond that, you can decide with your attorney if to register the mark with the U.S. Patent and Trademark Office for additional protection.