Doubts in understanding some concepts of potential energy
The potential energy of a system of point-like charges $q_i$ at positions ${\bf r}_i$ is $$ U=-\frac{1}{4 \pi \epsilon_0}\sum_i\sum_{j>i} \frac{q_i q_j}{|{\bf r}_i-{\bf r}_j|} $$
Such formula can be intepreted (and derived) as the work done by the Coulomb forces (better to avoid to introduce additional forces in the definition) to bring together the charges from an infinite relative distance, to their positions.
It turns out, looking at the formula, that this work can be interpreted as well as the sum of the work to assemble the first pair, summed to the work to add a third charge to the first pair, plus the work required to add a fourth particle to the first triple, plus ..., plus the work required to add the $N$-th charge to the previous $N-1$.
Does this observation allow to say that the energy of the system of $N$ charges coincides with the energy of one of the charges interacting with the other $N-1$ ?
Yes, because the previous formula says that. But one has to be careful to understand what is implicit in the formulae, if we would like to exploit them.
What has to be very clear with formula for potential energy is that in any case the potential energy remains a property of the whole system. This should be evident, if we think what would happen in a system of just two charges. We could fix one of them (say $q_1$) at its final position and then we evaluate the work done on the second charge (say $q_2$), when it is moved from infinity to its position. Even though we could speak of the work done by the force due to particle $1$ on particle $2$, and then speak about the potential energy of charge $q_2$, it is clear that it is a potential energy $U_{12}$ of the two-charge system. Indeed, if, after assembling the system, we fix charge $q_2$ and we free charge $q_1$, it starts to move according the to work-energy theorem, keeping fixed $K_1+U_{12}$, where $K_1$ is the kinetic energy of charge $q_1$.
Strictly speaking,
Potential energy of a charged particle at a point ( $\vec r $ ) is the amount of work done by the external force in bringing that charge from infinity to that particular point
Obviously, if there are no charges around (including static and in motion), the work done would be zero as the other charge would not experience any force.
Potential energy of a particular charge of the system ( q ) means you already had the other charges of your system already in place and then you bring the concerned charge q whose P.E. you want to find from infinity to that point.It can also be calculated by subtracting the potential energy of the system of other charges (excluding the charge q ) and subtracting it from the potential energy of the whole system ( including q )
Because potential energy can't be defined for a single charge, it is always defined for a system.
This is the issue. You can look at the energy contained in the system, or you can just look at a single charge $Q$. All you have to do is calculate.
$$U=-\frac{1}{4\pi\epsilon_0}\sum_i\frac{Qq_i}{r_i}$$ where we are summing over all charges except for the one in question. $r_i$ is the distance from charge $Q$ to charge $q_i$
I have been having some discussions in the comments of another answer about the validity of this. I want to address that I am not saying that potential energy is purely a property of a single body. I am also not saying that the idea of defining the potential energy contained in the entire system is incorrect. All I am saying is that it is not unreasonable to talk about the potential energy of a single body due to all of its interactions.
We typically do this in introductory physics when we use $mgy$ for the potential energy of an object in the nearly uniform gravitational field near the Earth's surface. Of course by using this we are not saying that the potential energy is only a property of the object of mass $m$. But I don't see an issue with saying the object has potential energy $mgy$ and therefore the force it experiences is $F=-\frac{\text d U}{\text d y}=-mg$.
Of course there is the issue mentioned in the comments of this answer of relating the potential energy of each charged object to the total potential energy of the system. There is a simple fix of just adding up each individual energy and then dividing by $2$ to address the double counting. I have seen this done in more than one text book.
I am not saying that this is the only "definition", or even that it is better than only considering things in terms of interactions. I am just trying to say that it is possible to use potential energy in this way. I think it works fine, and I don't think we should say we can't do this.