The Brooklyn City railroad was the original horsecar line which began operations in 1854. There were several other companies that came to be within the next decade. There were Horse car lines , a couple of cable car routes and finally electrification about 1895.
The BRT or Brooklyn Rapid Transit Corporation was formed around the turn of the century to consolidate transit in Brooklyn and Queens and Nassau under one corporate entity. There were but a few that did not become part of BRT most notable of which was the Manhattan Bridge 3 cent line running between Flatbush and Fulton st to Manhattan and canal street via the Manhattan Bridge.
The BRT had over its history a number of Labor Union diputes and strikes. During the 1919 strike, the well known Malbone street accident occurred which forced the company in receivership.
In 1923, the company was reorganized into the BMT or Brooklyn Manhattan Transit . It took over all of the BRT subway and elevated rail lines as well as the BRT trolley and bus lines that existed at the time . The Brooklyn and Queens Transit was organized as a separate company in 1929 and the BMT placed the operation of the Brooklyn and Queens Transit streetcar and bus routes at that time under B&QT Operation.
In June 1940, New York city organized the Board of Transport which acquired the assets and operations of the IRT and the BMT including the B&QT A process of unification began at that time. The City replaced the Board of Transport with the Transit Authority in 1953 and the first true unified line took place 1n 1954 with the BMT Culver line in Brooklyn on McDonald avenue, being re routed into the IND;s 6th avenue line at Church ave. All but three trolley lines in Brooklyn were gone by then , these last three closing by October 1956.
So to answer your question , All of the companies you noted were part of one company and the same company that oprated the streetcar lines in Brooklyn. From the Horse car of the Brooklyn City Railroad to the PCC car shown in the Transit Authority colors represents a span of 100 years of Streetcar lines in Brooklyn. A great post Bobby.