The Browning company -- which was started by John Moses Browning and his brothers -- has gone through several name changes over the years. This can be really confusing. Originally we were called Browning Brothers Company. Off and on we have been specifically called the Browning Arms Company. For nearly the last 50 years the business side of the company has operated under the simple name BROWNING. Other names, such as Browning Arms Company, have persisted in the common culture as the name of the company (such as on Wikipedia and business sites like Hoovers) and indeed have some significant business uses such as manufacturing and importing.
There are some abbreviations on products over the years, such as BACO or FN or Browning Viana, which may appear on some guns, but these are mostly for importing/manufacturing purposes. If your gun says it is made in Portugal, it is still a Browning. Browning and BACO are related, but not the exact same entity, but they clearly fall under the "Browning" umbrella of brands used somewhere in the process of making and selling Browning firearms. A number of firearms will have the actual words: Browning Arms Co. Morgan, Utah U.S.A. (such as found on the 1911-22 series) which is a correct inscription . . . the firearm IS made by Browning Arms Co. So, it is true that what your gun says and the name of our business may vary.
But when it comes to the company itself, Browning sporting arms and outdoor products are marketed and sold to dealers by simply the name Browning.