The name refers to the design of the display, regardless of the timekeeping technology used within the watch.
A digital watch is one in which the time is displayed as a series of digits, e.g. "3:27". An analog watch is one in which the display is not digital, but is indicated (typically) by the continuous motion of two or three rotating pointers or hands pointing to numbers arrayed on a circular dial (the hour hand's movement being analogous to the path of the sun across the sky).