Are the pitching moment and pitching moment coefficient of an airfoil given w.r.t. the aerodynamic center?
A moment is a free vector. There is no need to add a reference point - it can be shifted fore and aft or up and down without changing its effect. This effect is a rotation without translation.

So yes, as you suspect already, the pitching moment is a pure moment.

If there is also a force around (like on a wing), you have two vectors: One for the force and one for its moment. Put the reference point along the vector of the total force, and that moment disappears. Shift your angle of attack to the zero-lift point, and the force will disappear, leaving you with a pure moment (which is the sum of all the remaining forces along the chord which balance at this angle of attack).

In two-dimensional potential flow theory, all angle-of-attack-dependent forces act on the quarter point. If you now define your reference point at this quarter point, all variations in lift with angle of attack have no moment around this point! What is left is the pure pitching moment which does not depend on angle of attack.

If, however, you choose some other point as your reference point, the change in the lift force with angle of attack will add a moment, so your pitching moment is some constant value plus an angle-of-attack-dependent contribution. If you pick your reference point behind the quarter point on an airfoil with positive camber, there will be one angle of attack where those two moment components balance and the pitching moment will disappear. Then your reference point is the center of pressure at that angle of attack.

So the split of the aerodynamic forces into a force and a moment is purely arbitrary. Technically, we could always sum up all forces in the center of pressure, but then we need to keep track where it is, because it moves with angle of attack. Therefore, we decided to make things a bit more complicated and express the aerodynamic forces as one force at the quarter point and a separate moment. Why the quarter point? Because then this moment does not change with angle of attack. That is the advantage of defining all forces in the neutral point. We always know where the force acts at the price of carrying an extra moment with us. At least, this moment is constant over angle of attack - that is the big advantage of choosing the neutral point as the reference point.



EDIT

When looking for older answers to link to, I found that the two of us are discussing pitching moments since years without getting anywhere. Like here, here or here. I'm afraid I will repeat the same stuff like a broken record, and you keep asking the same question over and over again. Maybe it is better that you explain in a new question what your current understanding is so I can work out where exactly we two talk past each other.