"I think a few points of chemical background may help understanding the historical solutions to the dishwashing question.
With the exception of burnt residue (for which mechanical/abrasive treatment is most efficient), dish washing is about cleaning carbohydrates (starch/sugar), lipids (oil, grease) and protein from the dishes.
The carbohydrates are easiest: they are readily dissolved or emulsified by plain water, so no problem at all.
To get rid of the lipids, you can use an emulsifier (that term really just means any substance that helps form/keep lipid droplets in water). Soaps can do this, but also e.g. lecithin (remember reading: emulsifier soy lecitin - see the soybean powder in the comment. Although I doubt that historically edible substances were used much for dish washing - but some types of food already contain sufficient emulsifier so no additional emulsifier is needed for dish washing).
In alkaline/basic condiditons, fats (triglycerides) hydrolyze into glycerol and fatty acids which are deprotonated (anions) in the alkaline solution and thus have a hydrophilic end. These fatty acid anions act as emulsifier, so hydrolysis of a small part of the lipids is sufficient for dish washing.
Now, proteins that stick to the dish because they are denatured by heat (frying, baking - cooking also leads to heat denaturation, but due to being in water it doesn't stick to the dish as much) are the hardest problem for dishwashing. However, you can hydrolyze also these (for practical dishwashing: sufficiently to get the remainder off mechanically). Proteins hydrolyze easily in alkaline/basic conditions, but also in acidic conditions. Acidic hydrolysis is typically slower, though. A modern alternative is using enzymes.
Thus bases do help tremendously with dish washing. Two alkaline substances that have been around for very long time are
ashes (and their lye)
Wikipedia says about lye soap:
The ancient use of lye for soap-making and as a detergent is the origin of the English word, deriving from Proto-Germanic *laugo and ultimately from the Proto-Indo-European root *leue-, "to wash." Relatives in other Germanic languages, besides their words for lye, include the Scandinavian languages' words for Saturday (laugardagur, lördag, lørdag), meaning "washing day".
and lime
Plaster: There is archeological evidence that Pre-Pottery Neolithic B humans used limestone-based plaster for flooring and other uses.[12][13][14] Such Lime-ash floor remained in use until the late nineteenth century.
Lime (CaO) is not so good for dish washing, because its soaps [see below] are unsoluble in water, they form scum. This is why with hard water (containing much Ca²?) much more soap is needed than with soft water.
washing soda (Na2CO3) according to Wiki on dishwashing liquid was used before the modern dishwashing detergents were invented. It is the main component of natron, which was produced (among other uses also for washing and soap production) in ancient egypt.
Sodium carbonate became cheap with the invention of first Le Blanc and then the Solvay process, so we're talking 19th century here.
Soaps are the salts of fatty acids. They are produced by hydrolyzing fat in alkaline (NaOH or KOH) solution, e.g. lye from ash. For cleaning purposes, sodium (Na) and potassium (K) soaps are used. Being a salt of a strong base (KOH, NaOH) and a weak acid, they still react basic, so for dish washing they also have the property of inducing hydrolysis of protein in addition to being emulsifier for lipids.
Others have already given citations of ancient (Mesopotamia, Egypt etc.) knowledge of soap. Let me add that until sodium carbonate became a cheap industrial product, soap was a luxury article rather than a commodity (Ullmann's encyclopedia of industrial chemistry on Soap). From that I'd guess that the every-day dish washing was not done with anything so fancy*.
Also, not so ancient history from middle Europe (great-grandma's time, anecdata): Cast iron pans were wiped, but removing grease/fat was not even intended. If they were washed to remove grease, they were greased again immediately after to prevent rusting. This meant a taste to everything that we'd nowaday refuse as rancid but which was normal in former times (compare haut-goût). Again, burnt residue was and still is best removed with mechanical means: abrasives such as sand or steel wool.
Wooden plates were rinsed but also probably not washed to be totally grease-free. (Other cultures use one-way articles like banana leaves). In general, there was more cooking (and in general more soup/stew/porridge types of food) and less frying. And cooking pots are comparably easy to clean without (much) detergent if you start immediately or let them soak."
https://history.stackexchange.com/questions/45386/how-did-people-wash-dishes-before-dish-detergent