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?

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.

Potential energy does belong to the system. In particular, the potential energy of "an" object cannot be defined without reference to another. When we calculate, say, the gravitational potential energy "of a satellite", we compute

$$U = -G\frac{Mm}{r}$$

and two masses enter in: that of the Earth and that of the satellite. Moreover, both are interchangeable by commutativity, so it is just as valid to take the "focus" from the Earth - it just feels "weird" to us to imagine this, psychologically, since we think the active agent is the one that is dominant, but in truth the satellite is acting just as much - with just as much force - on the Earth as the Earth is acting on the satellite. The real difference is that the Earth is much more "lazy", or "inert", hence the term "inertia", than the satellite: it is vastly (on the order of $10^{21}$ if we take a 1 Mg satellite) less responsive to that force, but the truth is it feels just as strong a pull from the satellite itself.

Because both partners are equal participants, but not equally responsive to each other's mutual influence, and both enter into the calculation of potential energy, we consider it as shared between them. It would not exist with one partner missing - either one.

This becomes more evident when you get to more than two bodies: if you have three, you have to add up the potential from all 3 ways you can choose separate pairs of bodies. For $n$ bodies, you need to add up $\binom{n}{2}$ energies, not $n$. This number grows quadratically. This is also why the total potential energy of a solid sphere is proportional to the square of its mass (or charge, for electrostatics, which has the same mathematical form).

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