Alright, let us dive into some easy stuff first.
- What actually is potential energy?
Potential energy is defined as the energy possessed by a system by the virtue of it's configuration. This is the hard way. The easy way to understand is of course, building intuition, with some real life examples. Visualize this - you are an archer. Your bow string is at it's natural position, free of any strain. Suddenly, you decide to fire an arrow. You pull the string, and the Kinetic Energy of the string during this pulling gets totally converted into potential energy. When you've stretched it to the maximum, that is the situation where the energy of this system is only potential. But still, what is this energy?
Imagine the string to be at natural length. Now also imagine a 'dot' at this point. When you stretch this string, it goes farther and farther from the reference point, or the 'dot'. So basically, potential energy is the energy the string gains when it is pushed away from a reference point.
The mathematics now. Potential energy is mathematically defined as the negative of work done on a system.
$$ dU = -dW = -F.dr$$
$$ U_f - U_i = -W = -\int F.dr$$
Now, if you define $ U_i = 0 $ , that leaves you the potential energy of the system at any point away from the reference point, or the point where the energy was $U_i$. So, you see, you could define the surface of earth as the reference point, or the moon etc. Hope you get the point.
To your question now. The potential energy of a single object system depends on some things. Firstly, what kind of potential energy? Are you talking about the energy it has because of some charge lying next to it ( electrostatic energy)? Or due to a huge ball of rocks ( gravitational ) ? Are you saying that the object is lying around in vacuum, with nothing except the object there? Firstly, such a place is almost impossible, but still, if that be the case, that isolated piece doesn't actually have potential energy, because there is no force acting on it, which makes the integral above equal to zero. Hope this helps.