Break bulk cargoes (basically, cargo in boxes or other small units) was, pre-containerziation, moved by hand from the cargo hold to quayside warehouses or waiting trucks. If you disregard the increase in size of ships and treat trucks as equivalent to animal-drawn wagons, then yes, it was remarkably similar to how it had been.

However, other major cargo categorizes had been successful mechanized long before break-bulk was containerized. Tanker technology radically transformed how liquid cargos moved. (Up until tankers were developed, they were moved by barrel, and were basically a sub-category of break bulk.) Bulk cargoes (grain and ore) also had moved by a mix of barrel and crude small bulk sections which were hand loaded onto carts. The development of large scale bulk ships and the steam shovels and conveyor belts completely transformed bulk shipping and loading and unloading. Powered cranes also were essential in enabling the shipment of large, heavy equipment, which could be wrapped or boxed and simply hoisted on or off ships; before that, it was usually shipped in pieces (or, more often, not shipped at all, and locally manufactured).

Just to add a little more detail to /u/PlusGoody's well-informed post:

• The standard work on the history of containerization is Mark Levinson's The Box:How the Shipping Container Made the World Smaller and the World Economy Bigger - it's a fascinating read. See also, Brian Cudahy's Box Boats.

• The nineteenth century saw dramatic improvements in shipping technology, particularly through the use of machinery and steam power. Dockworkers had employed cranes to lift cargoes since the Middle Ages, but in 1838 Sir William Armstrong invented the hydraulic crane, which greatly accelerated the process of loading coal and other heavy cargoes. Five years later, Joseph Dart and Robert Dunbar invented the grain elevator, which used mechanical conveyors to load grain into ships, railway cars, and silos - essentially turning grain from a break bulk cargo packed in bags into a liquid cargo that was more easily stored and transported. Combined with the development of the tanker steamship in the 1870s-1880s, these innovations made the Victorian era a period of revolutionary change in merchant shipping - from 1870 to 1914 alone, maritime freight rates were cut in half because of improved productivity.