Angular momentum of an object is a physical quantity that depends on the chosen point about which to calculate the angular momentum.
It is often said that an object that has been thrown up in the air and is rotating, is physically rotating about the center of mass. I don't think that is what is physically happening. We chose the center of mass as the point of rotation because it is convenient mathematically (it makes the separation of translation and rotational energy easier). We could choose any other point (inside, outside the object, in motion or at rest relative to the object) and calculate the rotation about that arbitrary point.
It is not a physical fact that the free object rotates about the center of mass.
Even an object that is constrained to rotate about a fixed axis, we could describe the rotation about any point, not necessarily points on the fixed, constrained axis.