This dish soap contains sodium methyl 2-sulfolaurate/disodium 2-sulfolaurate. Sodium methyl 2-sulfolaurate/disodium 2-sulfolaurate is derived from coconut oil and acts as a surfactant and foaming agent. The chemical structure to the left of the sample is sodium methyl 2-sulfolaurate. The chemical structure to the right of the sample is disodium 2-sulfolaurate. In both cases, the anion head of the organic molecule (red in color) acts as a base and attracts the proton from water leaving behind a hydroxyl anion the excess of which creates a pH of 9 and turns the cabbage juice blue. The way the sulfolaurate anion generates OH- from water is similar to the way the hydrogen carbonate anion generated OH- anions with water in the baking soda example.
https://pubchem.ncbi.nlm.nih.gov/compound/Sodium-Methyl-2-Sulfolaurate