Why does the aerodynamic center exist?
It exists as a mathematical abstraction for conventionally shaped aerofoils at moderate speeds. In the end it is derived from the centre of pressure. The centre of pressure is the point where the moments of the individual aerodynamic forces over the aerofoil give 0 momentum. This always exist on any aerofoil. Now, the centre of pressure is not at a constant location, it is continually shifting as a function of angle of attack, and it moves quite a lot. However the rate of movement and change in lift force is in proportion to each other, and based on that, the stable point can be found, which for conventional aerofoils at normal speeds is about 25% chord.

Note, that to get this, the proportionality between lift and CP location is a must. You can create any shaped wing cross sections (might not want to call it an aerofoil) which will not have this property and won't necessarily have an aerodynamic centre. Whether it is fit to fly is questionable though. However, they will always have a CP, as it is a simple mathematical property, and it will very likely move with angle of attack. Also note, that the AC does change location with increasing speed, for supersonic speeds it moves significantly backwards.