The BBS was indeed started by a USFWS biologist, Chandler Robbins, in 1966. However it is presently a U.S. Geological Survey (USGS) project, run by the USGS Patuxent Wildlife Research Laboratory in Patuxent, Maryland, which also runs the Bird Banding Laboratory (BBL).
I'm not sure of the full story of how Patuxent came under management by the USGS, but one chapter of that story involved the former U.S. Biological Survey (or U.S. Biological Information Infrastructure). The Biological Survey was created in following a recommendation by the National Science Foundation in 1994, during the Clinton administration. The George W. Bush administration shut down the Biological Survey in the early 2000s and passed the BBS project to the USGS, as part of a government-reduction agenda. The current USGS website is rather opaque about the past history of both the Patuxent lab and the BBS, prior to the USGS taking responsibility.
It would certainly make sense for USFWS rather than USGS to be running both the BBS and the BBL. But I suppose that would require one undersecretary in the Department of the Interior to give up part of their fiefdom, and with it part of their annual budget allocation.