Based on this hundred-year-old advertisement, it looks like people used bar soap to wash dishes. According to Hunker, dishwashing detergents were invented during World War I and only came into common use during the mid-1900’s:
Soap was used for cleaning until 1916, when there was a shortage of fats needed to produce it during World War I. Because there was still a need for a cleaning product, synthetic versions were invented, which are now known as detergents.
There was also a movement towards using detergents because there was a need for a cleaning agent that did not leave behind a residue as soap did, especially on fabric. Upon their appearance, detergents became common products for cleaning dishes and clothing. While many people still used regular soap, by 1953 most households were using detergents.