I learned the total angular momentum is conserved in a two-body system. But as far as I saw the descriptions of two-body problem on several textbooks, none of those mention the angular momentum of each body. My question is: is the angular momentum of each body a conserved quantity?
The question requires an additional specification to allow a non-ambiguous answer, i.e., what is the reference frame used for describing the two-body problem. Indeed, if one would use the non-inertial reference frame centered on one of the two bodies, the angular momentum of that body would be trivially conserved (it remains zero forever), and the angular momentum of the other body will be conserved as a consequence of the central character of the gravitational force, and of the choice of the origin.
More interesting the case of an inertial frame. In such a frame, both bodies move, and there are two non-zero and non-conserved angular momenta. However, in the particular set of inertial frames with the origin coinciding with the center of mass, each angular momentum is constant, and of course, their sum is also constant.
The reason is that in the case of the classic gravitational two-body problem, the center of mass is always aligned with the line joining the two bodies. Therefore, the position vector, referred to the center of mass, is always aligned with the two force vectors. This implies a zero torque on each body, thus the conservation of each individual angular momentum. Notice that this is true only in such a special inertial frame.
Things go differently in the case of more than two bodies. In general, position vectors referred to the center of mass, and forces are not aligned. In the many-body case, only the total angular momentum is conserved.
No, the angular momentum of a closed system will always be conserved, but if you consider only one body within that system then its angular momentum will change based on its interactions with other bodies in the system just like linear momentum.
Physics problems usually require a lot of simplification of the system which results in point object assumption. No object is really a true point, but if there were a real point object then it will not have any spin as it will be impossible to distinguish the rotational positions.
So when you look at textbooks with point body problems they will not have any reason to consider rotational mechanics. But if you consider real-world systems like Earth and Moon then the conservation of Angular momentum plays a very important role in deciphering the rotational velocities and the phenomenon of tidal locking.