newtonian mechanics - Does potential energy belong to an object or the system?

The gravitational potential energy of an object is given by mgh where h is the distance from the chosen zero potential reference level.

Usually we talk as if the potential energy belong to the object although it belongs to the system (eg. the Earth and object) as I understand it. Is the reason we speak like that because the Earth has so much more mass than the object, and gravitational field around isn't affected by the mass?

Is it perfectly okay to state that the object 'have' or 'possess' potential energy?

Answer

Usually we talk as if the potential energy belongs to the object although it belongs to the system (eg. the Earth and object) as I understand it. Is the reason we speak like that because the Earth has so much more mass than the object...?

Yes, that is correct. For a system of two bodies interacting with each other gravitationally, like the earth and a pebble, the total energy of the system is $$ K_\text{earth} + K_\text{pebble} + V(r) $$ where $K_\text{earth}$ and $K_\text{pebble}$ are the kinetic energies of the objects and where $V(r)$ is the potential energy, which depends on the distance $r$ between the centers of the two objects. The potential energy depends on $r$, and $r$ depends on the locations of both objects (earth and pebble), so the potential energy is a property of the system, not a property of either object individually. But, like you said, since the mass of the earth is so much greater than the mass of the pebble, the pebble will have a negligible effect on the earth. In this case, we can use an approximation in which the earth remains at a fixed location — and then $r$, and therefore $V(r)$, might as well just be a property of the pebble. This is only an approximation, but it is an excellent approximation.

In general, energy is a property of the whole system, not of the individual parts of the system. Sometimes we can use approximations in which the potential energy can be regarded as a property of just one object; but in general, energy is only a property of the whole system. Only the total energy of the whole system is convserved, and the fact that it is conserved is the reason it is important.