no they dont really have more hydrogen ions, acids and bases are not defined by whether they do or dont have hydrogen ions, but by HOW they interact with other substances. The Bronsted-Lowry definition of an acid is a proton donor. (A proton is a hydrogen ion). In a solution, the hydrogen ions in an acid dissociate to produce hydronium ions. So as the acids are "donating" hydrogen ions, the concentration of hydrogen ions in an acidic solution is higher. acids donate protons to bases in reactions
hope this makes sense