There is an aerodynamic damping effect in yaw just as there is in roll. Imagine that the whole aircraft has somehow become yawed 10 degrees to the flight path, and then whatever torque has caused this to happen has vanished. The fin will generate a yaw torque that makes the plane start to yaw back toward the flight path, but as it yaws, the rotational velocity changes the local linear velocity of different points on the fuselage, which changes in the "sideways angle-of-attack" of the forward parts of the fuselage in a way that decreases the net yaw torque on the aircraft. Specifically, the yaw rotation will increase the "sideways angle-of-attack" of the forward parts of the aircraft and decrease the "sideways angle-of-attack" of the aft parts of the aircraft, including the vertical fin. At some high enough rotation rate, the net yaw torque would be zero even though the fin is still "feeling" a sideways component in the airflow.
One way to envision is this is to understand that in steady turning flight, whenever the aircraft's yaw and pitch rotation rates are stable and fully synchronized to the turn rate, we can envision the "relative wind" or free-stream airflow to actually be curved to follow the path of the turn, so that different points on the fuselage cannot all have zero "sideways" angle-of-attack, nor can they all have the identical pitch-wise angle-of-attack.
In short, the rotation of the aircraft in the yaw axis causes different parts of the aircraft to be moving through the airmass in different directions at any given instant in time, so that "yaw strings" attached to different points on the aircraft would be angled in slightly different directions. To read about how one pilot takes this into account in actual practice, see the Soaring magazine article Circling the Holghaus way by Richard H. Johnson. Links to some other articles discussing the curvature of the relative wind in turning flight are embedded in this ASE answer.
As for your question about the "weathervane effect" in the pitch axis, you could say that the balancing act between the wing and the horizontal tail is sort of like a "weathervane effect". Airfoils generally generate a nose-down pitching moment. Since apparent "center of lift" of the wing (including the effect of this nose-down pitching moment) is usually not at the CG of the aircraft, the point where everything comes into balance will usually not be the point where the horizontal tail is generating exactly zero lift.
Naturally, there is an aerodynamic damping effect for pitch rotations as well. The difference in angle-of-attack between the wing and the tail required to command a given angle-of-attack of the wing in turning flight will be different than in linear flight, especially at steep bank angles where the turn mostly involves pitch rotation. Again this is due to the fact the pitch rotation creates a local change in angle-of-attack that varies from one point along the fuselage to another. Generally we have to have the stick further aft to command a given angle-of-attack of the wing in turning flight than in linear flight. This is effect is especially pronounced in slow-flying aircraft like sailplanes. If the fuselage could "bend like a banana" to match the curving relative wind in the turn, this effect would vanish.
The idea of a "curvature" in the free-stream "relative wind" is really just another way to look at aerodynamic damping due to rotation about the pitch or yaw axes, while aerodynamic damping due to rotation about the roll axis can be viewed as a being due to a "twist" in the free-stream "relative wind".
