Alright, so maybe we're not here to talk about your dumpster exactly, but the dumpster, per se. The large, sturdy commercial dumpster, as we know it today, was the invention of the Dempster brothers, patented in 1935. So, the dumpster is actually the product of the Great Depression, when homes were being foreclosed on, which meant debris to cart away, and at the same time the US government was creating public works projects that resulted in a number of construction sites that needed some heavy duty dumpsters.
The word "dump" was around prior to the Dempster's invention, but they thought it would be clever to combine their name with the verb, as they primarily designed their containers to fit on the back of trucks that hauled debris from one place and dumped it in another. The Dempster Dumpmaster was the first front-loading garbage truck and popularized both the use of the term "dumptruck" and the use of "dumpster."
The reason your trusty local dumpster rental company, such as Dumpsters Unlimited, can use the word "dumpster" in their name is because the trademark has expired. However, the mission of the Dempster brothers lives on in cities and municipalities throughout the world.