The thickening and setting of the meringue is due to protein denaturation and coagulation of the ovalbumin in the egg whites. First, the protein is agitated by beating air into the mixture, then sugar is added, which dilutes the foam. This creates a looser coagulation with more structure that makes it easier to catch air bubbles making it bigger and stiffer. The mixture is set by heat when it is cooked in the double boiler and baked, which changes the ovalbumin to S-ovalbumin and solidifies the structure with the air bubbles trapped inside.
There is no thickening due to starch because there are no starch granules and no mixing of fat and water phases because there is no fat phase.